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MATHS LECTURE # 09
Plane Geometry
Point, line and plane A line segment is a part of a line between two endpoints. It is named by its endpoints, for example, A and B. There are three basic concepts in geometry. These concepts are the “point”, “line” and “plane”. A Point B AB is a line segment. It has a definite length. A fine dot, made by a sharp pencil on a sheet of paper, resembles a geometrical point very closely. The sharper the pencil, the closer is the dot to the concept of a geometrical point. Plane The surface of a smooth wall or the surface of a sheet of paper or the surface of a smooth blackboard are close examples of a plane. The surface of a blackboard is, however, limited in extent and so are the surfaces of a wall and a sheet of paper but the geometrical plane extends endlessly in all directions. If point P is on the line and at the same distance from A as well as from B, then P is the midpoint of segment AB. When we say AP = PB, we mean that the two line segments have the same length. A B P A part of a line with one endpoint is called a ray. AC is a ray of which A is an endpoint. The ray extends infinitely far in the direction away from the endpoint. Lines A line in geometry is always assumed to be a straight line. It extends infinitely far in both directions. It is determined if two points are known. It can be expressed in terms of the two points, w h i c h a r e w r i t t e n i n c a p i t a l l e t t e r s . T h e f o l l o w i n g l i n e i s c a l l e d A B . C Angles An angle is formed by two rays with the same initial point. So an angle looks like the one in the figure below. B C Or, a line may be given one name with a small letter. The following line is named line k. A k B Definition: An angle is the union of two non­collinear rays with a common initial point.
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IC : PTtkmml09 (1) of (20) The two rays forming an angle are called the “arms” of the angle and the common initial point is called the “vertex” of the angle. Notation: The angle formed by the two rays AB and AC, is denoted by the symbol ÐBAC or ÐCAB. Supplementary angles If the sum of two angles is a straight angle (180°), the angles are supplementary and each angle is the supplement of the other. It is at times convenient to refer the angle BAC, simply as ÐA. However this cannot be done if there are more than one angle, with the same vertex A. Also, it is convenient to denote angles by symbols such as
Ð1, Ð2, Ð3, etc. E a
b G Ð G is a straight angle = 180°
D \ Ð a + Ð b = 180°.
Þ Ð a and Ð b are supplementary angles. 3 2 A C E1. What is the supplement of an angle whose measure is 57°? 1 B Types of angles Sol. The supplement = 180° – 57° = 123°. Vertical angles E2. The supplement of an angle is one third of the angle. Find the supplement of the angle. When two lines intersect, four angles are formed. The angles opposite to each other are called vertical angles and are equal to each other. Sol. Let the angle be x. So, the supplement is x/3. Now, x + x/3 = 180° or 4x/3 = 180° or x = 135°. Hence, the supplement = 135/3 = 45°. d a c b Right angles a and c are vertical angles, Ð a = Ð c. If two supplementary angles are equal, they are both right angles. A right angle is one­half of 180°. Its measure is 90°. A b and d are vertical angles, Ð b = Ð d. right angle is symbolised by Straight Angle . ÐG is a straight angle.
A straight angle has its sides lying along a straight line. It is always equal to 180°. A B C
K
Ðb + Ða = ÐG and Ða = Ðb.
\ Ða and Ðb are right angles. Complementary angles b a G If the sum of two angles is 90°, the angles are complementary and each angle is the complement of the other. Ð ABC = Ð B = 180°. i.e., Ð B is a straight angle. Adjacent angles Two angles are adjacent if they share the same vertex and a co mmon si de but no angl e i s i nsid e t he o the r a ngle . Ð AB C and ÐCBD are adjacent angles. Even though they share a common vertex B and a common side AB, Ð ABD and Ð ABC are not adjacent angles because one angle is inside the other. Y b a For example, if Ð Y is a right angle and
Ð a + Ð b = Ð Y = 90°, then
Ð a and Ð b are complementary angles. A E3. What is the complement of an angle whose measure is 47°? C Sol. The complement = 90° – 47° = 43°.
B (2) of (20) D IC : PTtkmml09 E4. The complement of an angle is two third of the angle. Find the complement of the angle. Sol. Let the angle be x. So, the complement is 2x/3. Now, x + 2x/3 = 90° or 5x/3 = 90° or x = 54°. 54 ´ 2 = 36° . Hence, the complement = 3 Some properties of angles Linear pair of angles From the figure Ð AOC and Ð BOC are adjacent angles. Look at the non­common arms, OA and OB. These are two opposite rays or these are collinear. We have a name for them also. Acute angles C Acute angles are the angles whose measure is less than 90°. No two acute angles can be supplementary. Two acute angles can be complementary angles. A O B Definition: Two adjacent angles are said to form a linear pair of angles if their non­common arms are two opposite rays. Now, look at the figure below C or C
C Ð C is an acute angle. C B Obtuse angles A Obtuse angles are the angles whose measure is greater than 90° and less than 180°. O A (i) B (ii) In each of the figures above, Ð AOC and Ð COB form a pair of adjacent angles. In figure (ii) we have a linear pair of angles. If adjacent angles Ð AOC and Ð COB form a linear pair, then
or
D O D Ð AOC + Ð COB = 180°. Ð D is an obtuse angle. Imp. Reflex angle An angle with measure more than 180° and less than 360° is called a reflex angle. Two adjacent angles are a linear pair of angles if and only if they are supplementary. Parallel lines a 180°< a < 360° Two lines meet or intersect if there is one point that is on both lines. Two different lines may either intersect at a single point or never intersect, but they can never meet in more than one point. a E5. Categorise the following angles as acute, right, obtuse, reflex and straight angle : 37°, 49°, 138°, 90°, 180°, 89°, 91°, 112°, 45°, 235°. Sol. Acute angles : 37°, 49°, 45°, 89°. Right angle : 90°. b P Two lines in the same plane that never meet no matter how far they are extended are said to be parallel, for which the symbol is ||. In the following diagram a || b. Obtuse angles : 138°, 91°, 112°. Reflex angle : 235°. a Straight angle : 180°. b
IC : PTtkmml09 (3) of (20) Imp. Two lines intersected by a transversal are parallel, if Ø If two lines in the same plane are parallel to a third line, 1 . the corresponding angles are equal, or then they are parallel to each other. Since a || b and b || c, we know that a || c. 2 . the alternate interior angles are equal, or 3 . the alternate exterior angles are equal, or 4 . interior angles on the same side of the transversal are supplementary. a b c Ø y x
z w Line a Two lines that meet each other at right angle are said to be per pendi cular, for whi ch th e sym bol i s ^ . Line a is perpendicular to line b. q p r s Line b a If a || b, then 90° 90° b Ø Two lines in the same plane that are perpendicular to the 1.
Ðy = Ðq. Ðz = Ðr, Ðx = Ðp and Ðw = Ðs. 2.
Ðz = Ðp and Ðw = Ðq. 3.
Ðy = Ðs and Ðx = Ðr. 4.
Ðz + Ðq = 180° and Ðp + Ðw = 180°. same line are parallel to each other. a 90° Since vertical angles are equal, \ Ðp = Ðr, Ðq = Ðs, Ðy = Ðw and
Ðx = Ðz. If any one of the four conditions for equality of angles holds true, then the lines are parallel. b 90° c Ø E6. In the given figure, two parallel lines are intersected by a transversal. Find the measure of angle y. Line ‘a’ ^ line ‘c’ and line ‘b’ ^ line ‘c’. \ a || b. A line intersecting two other lines is called a transversal. Line ‘c’ is a transversal intersecting lines ‘a’ and ‘b’. a c 2x 3x+50° y b Sol. The two labelled angles are supplementary.
\ 2x + (3x + 50°) = 180°. The four angles between the given lines are called interior angles and the four angles outside the given lines are called exterior angles. If two angles are on opposite sides of the transversal, they are called alternate angles. 5x = 130° Þ x = 26°. S ince Ðy is vertical to the angle whose measurement is 3x + 50 °, it has the same measurement.
\ y = 3x + 50° = 3(26°) + 50° = 128°.
y x
z w q p r s
Ðz, Ðw, Ðq and Ðp are interior angles.
Ðy, Ðx, Ðr and Ðs are exterior angles.
Ðz and Ðp are alternate interior angles, so are Ðw and Ðq.
Ðy and Ðs are alternate exterior angles, so are Ðx and Ðr. Pairs of corresponding angles are Ðy and Ðq; Ðz and Ðr; Ðx and
Ðp and Ðw and Ðs. (4) of (20) IC : PTtkmml09 Mini Revision Test # 01 Chord of a circle is a straight line that divides the circle into two parts. DIRECTIONS: Fill in the blanks. In case of the diameter, it is the largest possible chord that can be drawn in a circle. 1 . A part of a line with one end point is called a..... Imp. 2 . Two adjacent angles are a linear pair of angles if and only if they are........ 3 . A line intersecting two other lines is called a..... 4 . Two angles are said to be supplementary if their sum is ..... 5 . Two angles are said to be complementary if their sum is ..... 6 . The angle formed by the two rays AB and AC, is denoted by the symbol ....... or ....... 7. When two lines intersect, four angles are formed. The angles opposite each other are called ....... 8 . Two angles are ....... if they share the same vertex and a common side but no angle is inside another angle. 9 . If a ray stands on a line, the sum of the two adjacent angles so formed is ....... 10. Two lines in the same plane that are perpendicular to the same line are ...... to each other. The problem of finding lengths and areas related to figures with curved boundaries is of practical importance in our daily life. I f A an d C d en o te t h e ar ea an d pe r im et er (a ls o c al le d circumference) respectively, of a circle of radius r, then A = pr 2 , C = 2pr Here p (a letter of Greek alphabet) is the ratio of circumference to the diameter of a circle. It stands for a particular irrational number whose value is given, approximately to two decimal places, by 3.14 or by the fraction 22 . 7 E7. Find the circumference of a circle with area 25p sq ft. Sol. Area of the circle = pr 2 = 25p.
\ r = 5 ft.
\ circumference = 2pr = 10p ft.
Circle E8. Find the area of a circle with circumference 30p cm. The following figure shows the parts of a circle. Sol. Circumference = 2pr = 30p
r = 15 cm.
Diameter \ area = pr 2 = p × 15 2 = 225p sq. cm. s diu
Ra
Sector and segment of a circle: See the figures given below. Centre C Circle Centre of a circle is the point within that circle from which all straight lines drawn to the circumference are equal in length to one another. O q
Radius of a circle is the distance from the centre of the circle to the circumference. Diameter of a circle is a line that runs from the edge of the circle, through the centre, to the edge on the other side. The diameter is twice the length of the radius. IC : PTtkmml09 O B A
P Q R (i) Circumference of a circle is the outer periphery of the circle. It is also known as the perimeter of the circle. S (ii) In figure (i) note that the area enclosed by a circle is divided into two regions, viz., OBA (shaded) and OBCA (non­shaded). These regions are called sectors. A minor sector has an angle q, subtended at the centre of the circle. The boundary of a sector consists of arc of the circle and the two lines OA and OB. In fig (ii), the area enclosed by a circle is divided into two segments. Here segment PRQ (shaded) is the minor one and segment PQS (non­shaded) is the major one. The boundary of a segment consists of an arc of the circle and the chord dividing the circle into segments.
(5) of (20) Area of a sector and length of arc Ø The arc length l and the area A of a sector of angle q in a circle of radius r, are given by Ø l =
2 1 pr q
pr q
. We also note that A = lr . , A =
2 180 360 In a circle with centre O, ÐACB and ÐADB are the angles at the circumference, by the same arc AB, then ÐACB = ÐADB. (Ðs in same arc or Ðs in same seg.) In a circle with centre O and diameter AB, if C is any point at the circumference, then ÐACB = 90°. C Theorems on Circles Ø O A If ON is ^ from the centre O of a circle to a chord AB, then AN = NB. B (Ð in semi­circle) O A Ø If ÐAPB = ÐAQB and if P, Q are on the same side of AB, then A, B, Q, P are concyclic. B N P (^ from centre bisects chord) Q Ø If N is the midpoint of a chord AB of a circle with centre O, then ÐONA = 90°. (Converse, ^ from centre bisects chord) Ø Equal chords in a circle are equidistant from the centre. If the chords AB and CD of a circle are equal and if OX ^ AB, OY ^ CD, then OX = OY. A B (AB subtends equal Ðs at P and Q on the same side) Ø A C O In equal circles (or in the same circle) if two arcs subtend equal angles at the centre (or at the circumference), then the arcs are equal. P O X Q Y O B B D (Eq. chords are equidistant from centre) Ø Ø If OX ^ chords AB, OY ^ chord CD and OX = OY, then chords AB = CD (Refer the figure given above). (Chords equidistant from centre are eq.) A circle with centre O, with ÐAOB at the centre, ÐACB at the circumference, standing on the same arc AB, then
ÐAOB = 2ÐACB. C A X If ÐBOA = ÐXOY, then AB = XY or if f ÐBPA = ÐXQY, then AB = XY . (Eq. Ðs at centre or at circumference stand on equal arcs) In the same or equal circles if chords AB = CD, then arc AB = arc CD. || B || D A O B D A (Ð at centre = 2Ð at Y O C (Eq. chords cut off equal arcs)
Circumference on the same arc AB) (6) of (20) IC : PTtkmml09 Few important results Ø E10. The length of the minute hand of a clock is 7 cm. Find the area swept by the minute hand in 10 minutes. (Take p = 22/7). If two chords AB & CD intersect externally at P then Sol. The minute hand describes a circle of radius 7 cm. It describes an angle of 360 ° = 6° per minute. 60 A B P Thus, we get a sector of angle 6° and radius 7 cm per minute. D Its area
C
\ Area of the sector described in one minute = 2.567 sq. cm.
+ PA × PB = PC × PD
+ ÐP = 1 2 [m (arc AC) – m (arc BD)] Ø pr 2 q 22 7 ´ 7 ´ 6 77
=
´
=
= 2.567 sq. cm.
360 °
7 360 30
Þ Area described in 10 minutes = 2.567 × 10 sq. cm = 25.67 sq. cm. E11. A sector is cut from a circle of radius 21 cm. The angle of the sector is 150°. Find the length of its arc and area. If PAB is a secant and PT is a tangent then,
Sol. Length of the arc of the sector, B A l =
P Y X
=
T 2
=
E9. Complete the following table. 150 2
´ pr sq. cm.
360 150 22 2 ´
´ 21 ´ 21 cm = 577.5 sq. cm. 360 7 Mini Revision Test # 02 Radius r r Area A Circumference C 1 1 ­­ ­­ 2 4 ­­ ­­ 3 9 ­­ ­­ 4 16 ­­ ­­ Sol. Radius r r Area = pr C = 2pr 1 1
p
2p
2 4 4p
4p
3 9 9p
6p
4 16 16p
8p
2 150 22 ´
´ 21 cm = 55 cm. 180 7 Area of the sector =
+ PA × PB = PT + ÐP = 1/2[m(arc BXT) – m(arc AYT)] 2 150 ´ pr cm.
180
2 DIRECTIONS: Determine whether the given statements are TRUE or FALSE. 1 . The sum of the interior angles of a pentagon is 540°. 2 . The perimeter of a regular hexagon with side a is 6a. 3 . The circumference of a circle whose radius is 14 cm is 176 cm. 4 . The perimeter of a regular hexagon with side 8 cm is 48 cm. 5 . The area of a quadrant of a circle with radius 14 cm is 148 sq. cm. DIRECTIONS: Fill in the blanks. 6 . All circles measure .... degrees. 7. The area of a regular hexagon with side 8 cm is ..... 8 . If the radius of a circle is decreased by 50%, its area will decrease by .... 9 . If the diameter of a circle is increased by 100%, its area is increased by ... 10. The length of minute hand on a wall clock is 14 cm. The area swept by it in 30 minutes is ...
IC : PTtkmml09 (7) of (20) Polygon Definition: A polygon is a closed figure bounded by straight lines. Each of the line­segments forming the polygon is called its side. The angle determined by two sides meeting at a vertex is called an angle of the polygon. A When two sides of the polygons are equal, we indicate the fact by drawing the same number of short lines through the equal sides. E A F B A D G C B E B C C A D This indicates that AB = EF and CD = GH. Two polygons with equal corresponding angles and corresponding sides in proportion are said to be similar. The symbol for similar is ~. F B A D E H ~ E B F C G Similar figures have the same shape but not necessarily the same size. D C H Remark: For n = 3, the polygon has the special name­ triangle and similarly for n = 4, the polygon has the special name­ quadrilateral. Some other special names are Pentagon (n = 5). Triangles and their types Hexagon (n = 6). Definition: A polygon P 1 P 2 .... P n is called convex if for each side of the polygon, the line containing that side has all the other vertices on the same side of it. All the three figures given (triangle, pentagon and hexagon) are convex polygons. A triangle is a polygon with three sides. Triangles are classified by measuring their sides and angles. The sum of the angles of a plane triangle is always 180°. The symbol for a triangle is D. The sum of any two sides of a triangle is always greater than the third side. Equilateral Regular polygon: A polygon is called a regular polygon if all its sides and angles are equal. In the given figure, the hexagon is a regular polygon. Equilateral triangles have equal sides and equal angles. Each angle measures 60°. A In the D ABC, AB = AC = BC.
Congruent and similar polygons \ ÐA = ÐB = ÐC = 60°. If two polygons have equal corresponding angles and equal corresponding sides, they are said to be congruent. Congruent polygons have the same size and shape. The symbol for congruence is @.
A
D E H F G B C Isosceles Isosceles triangles have two sides equal. The angles opposite to the equal sides are equal. The two equal angles are sometimes called the base angles and the third angle is called the vertex angle. Note that an equilateral triangle is isosceles. @
B C F In the DFGH, FG = FH. FG ¹ GH.
ÐG = ÐH.
G H ÐF is vertex angle.
ÐG and ÐH are base angles.
(8) of (20) IC : PTtkmml09 Scalene Properties of triangles Scalene triangles have all three sides of different length and all angles of different measure. In a scalene triangle, the shortest side is opposite the angle of smallest measure and the longest side is opposite the angle of greatest measure. Ø Ø Sum of the three interior angles of a triangle is 180°. Ø Ø Interior angle + corresponding exterior angle = 180°. Ø Ø Sum of any two sides is always greater than the third side. In the DABC, AB > BC > CA.
\ ÐC > ÐA > ÐB. A When one side is extended in any direction, an angle is formed with another side. This is called the exterior angle. An exterior angle = Sum of the other two interior angles not adjacent to it. A triangle must have at least two acute angles. Properties of similar triangles B C If two triangles are similar, then ratio of sides = ratio of heights = ratio of perimeters = ratio of medians = ratio of angle bisectors = ratio of inradii = ratio of circumradii. Right angled triangle Right angled triangle contains one right angle. Since the right angle is 90°, the other two angles are complementary. They may or may not be equal to each other. The side of a right triangle opposite the right angle is called the hypotenuse. The other two sides are called legs. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. In the above result, ratio of sides refers to the ratio of corresponding sides etc. Also, ratio of areas = ratio of squares of corresponding sides. Right triangle A In the DABC, AC is the hypotenuse. 90° A AB and BC are legs.
ÐB = 90 °.
c b B \ ÐA + ÐC = 90°.
Þ
a 2 + c 2 = b 2 . 90° D C ABC is a right triangle with ÐA = 90°. B a C If AD is perpendicular to BC, then E12. ABC is a right triangle with ÐB = 90°. If AB = 12 and BC = 5, then what is the length of AC? Sol. AB 2 + BC 2 = AC 2 .
(a) Triangle ABD ~ Triangle CBA and BA 2 = BC × BD. (b) Triangle ACD ~ Triangle BCA and CA 2 = CB × CD. (c) Triangle ABD ~ Triangle CAD and DA 2 = DB × DC. Þ AC 2 = 12 2 + 5 2 = 144 + 25 = 169.
\ AC = 13. E13. Show that a triangle with sides 15, 8 and 17 is a right triangle. Sol. The triangle will be a right triangle if a 2 + b 2 = c 2 .(as shown above)
Two triangles are similar Two triangles are similar if the angles of one of the triangles are r e s p ec t i v e l y e q u a l t o t h e a n g l es o f t h e o t h e r a nd t h e corresponding sides are proportional.
\ 15 2 + 8 2 = 17 2 .
Þ 225 + 64 = 289.
\ the triangle is a right triangle and 17 is the length of the hypotenuse. IC : PTtkmml09 (9) of (20) (Angle . side . angle) A.S.A. Theorems If ÐB = ÐY, BC = YZ, ÐC = ÐZ, then triangle ABC @ triangle XYZ. (Equiangular triangles) A.A.A. If two triangles are equiangular, their corresponding sides are proportional. In triangles ABC and XYZ, if ÐA = ÐX,
(Side . side . side) S.S.S. If AB = XY, AC = XZ, BC = YZ, triangle ABC @ triangle XYZ. ÐB = ÐY, ÐC = ÐZ, then AB/XY = AC/XZ = BC/YZ. (Right angle hypotenuse side) (R.H.S) A B X C Y A X Z B (3 sides proportional) If 2 triangles have their corresponding sides proportional, then they are equiangular. In triangles ABC and XYZ, if AB/XY = AC/XZ = BC/YZ, then ÐA =ÐX, ÐB =ÐY, ÐC =ÐZ. C Y Z If ÐB = ÐY = 90°, Hypotenuses AC = XZ, AB = XY, then triangle ABC @ triangle XYZ. Note: Congruent triangles are definitely similar but similar triangles may not be congruent. (Ratio of two sides and included angle) If one angle of the triangle is equal to one of the angles of the other triangle and the sides containing these angles are proportional, then the triangles are similar. If ÐA = ÐX and AB/XY = AC/XZ, Mini Revision Test # 03 DIRECTIONS: Answer the questions. then ÐB = ÐY, ÐC = ÐZ. 1 . If triangles ABC and XYZ are similar, then it is denoted by
DABC ~ DXYZ. What is the measure of each acute angle of an isosceles right angled triangle? 2 . One of the acute angles of a right triangle is 42°. Find the measure of the other acute angle. Congruency of triangles 3 . Two angles of a triangle measure 92° and 58° respectively. Find the measure of the third angle. Two triangles are said to be ‘CONGRUENT’ if they are equal in all respects (sides and angles). 4 . One of the acute angles of a right triangle is double the other. Find the measure of its smallest angle. 5 . Each of the equal angles of an isosceles triangle is double the third angle. What is the measure of the third angle? 6 . The measures of the angles of a triangle are in the ratio 2 : 3 : 4. Find the measure of its greatest angle. The three sides of one of the triangle must be equal to the three sides of the other, respectively. 7. One of the angles of a triangle is equal to the sum of the other two. Find the measure of the greatest angle. The three angles of the first triangle must be equal to the three angles of the other respectively. Thus, if D ABC and D XYZ are congruent, (represented as D ABC @ D XYZ), then 8 . What is the measure of each angle of an equilateral triangle? 9 . The measures of the acute angles of a right triangle are in the ratio 1 : 5. Find the measure of its angles. A B X C Y Z AB = XY, AC = XZ, BC = YZ and
ÐA = ÐX, ÐB = ÐY, ÐC = ÐZ. Theorems (Side . angle . side) S.A.S. If AB = XY, ÐA = ÐX, AC = XZ, then triangle ABC @ triangle XYZ. (Angle . angle . side) A.A.S. 10. Match the following i. Right triangle ii. Isosceles right triangle ii. i. 60°, 60°, 60°. iii. Obtuse scalene triangle iii. 90°, 42°, 48°. iv. Equilateral triangle 108°, 36°, 36°. v. Acute angled isosceles v. iv. 120°, 28°, 32°. 45°, 45°, 90°. triangle vi. Obtuse isosceles triangle vi. 65°, 50°, 65°.
If ÐB = ÐY, ÐC = ÐZ, AC = XZ, then triangle ABC @ triangle XYZ. (10) of (20) IC : PTtkmml09 Square Quadrilaterals A A quadrilateral is a polygon of four sides. The sum of the interior angles of a quadrilateral is 360°. D a a B Parallelogram Some important properties D A Ø
Ø P B C C ÐA = ÐB = ÐC = ÐD = 90°. AB = BC = CD = DA. Trapezium (Trapezoid) AD || BC
ÐA + ÐB = 180° AD = BC
D ABD @ D CDB AB || DC
D ABC @ D CDA AB = DC AP = PC
ÐD = ÐB BP = PD
b 1 A AD || BC. AD and BC are bases. h AB and DC are legs. B h = altitude. Imp. Rhombus A D a If in a parallelogram, all the sides are equal, then the parallelogram so formed is a rhombus. a P C B If a quadrilateral is not a parallelogram or a trapezoid but it is irregularly shaped, its area can be found by dividing it into triangles, attempting to find the area of each and adding the results. The chart below shows at a glance the properties of each type of quadrilateral. Some important properties ÐDAB = ÐDCB & ÐADC = ÐABC. Quadrilateral AP = PC, DP = PB & ÐAPD = 90°. ÐADP = ÐPDC, ÐABD = ÐPBC. ÐADC + ÐDCB = 180° & ÐDAB + ÐABC = 180°. Rectangle A D P
B b C l Some important properties Ø Ø C b 2 ÐA = ÐC Ø
Ø Ø
Ø
D Parallelogram Trapezium (Opposite sides and opposite (One pair of opposite sides are parallel) angles are equal. Diagonals bisect each other)
Rectangle Opposite sides are equal. Each angle is 90°. Diagonals bisect each other. Diagonals are of equal length. Square Rhombus All sides are All sides are equal. Each angle equal. Opposite is 90°. Diagonals angles are equal. bisect each other Diagonals bisect at right angles. each other at Diagonals are right angles. equal. AD = BC, DC = AB, ÐA = ÐB = ÐC = ÐD = 90°. DP = PB = AP = PC. IC : PTtkmml09 (11) of (20) Ø Measure of an arc means measure of central angle. Mini Revision Test # 04 DIRECTIONS: Fill in the blanks. m(minor arc) + m(major arc) = 360°. Ø Angle in a semicircle is a right angle. Ø Ø The sum of all the angles of a triangle is 180°. Ø Exterior angle of a triangle is equal to the sum of opposite interior angles. Ø Conditions of congruence of triangles are SAS, SSS, ASA and RHS. Ø Angles opposite to equal sides in a triangle are equal and vice versa. Ø If two sides of a triangle are unequal, the greater side has greater angle opposite to it and vice versa. Ø If the square of the longest side of a triangle is equal to the sum of squares of the other two sides, the triangle is right­ angled. Angle opposite to the longest side is a right angle. Ø If the square of the longest side of a triangle is greater than the sum of squares of the other two sides, the triangle is obtuse­angled. Angle opposite to the longest side is obtuse. Ø If the square of the longest side of a triangle is less than the sum of squares of the other two sides, the triangle is acute­angled. Ø Perimeter of a triangle is greater than the sum of its medians. Ø Triangles on the same or congruent bases and of the same altitude are equal in area. Ø Triangles equal in area and on the same or congruent bases are of the same altitude. Ø Parallelograms have opposite sides and opposite angles equal. + Important results and formulae Ø Diagonal of a parallelogram divides it into two congruent triangles. Ø Only one circle can pass through three given points. Ø There is one and only one tangent to the circle passing Ø Diagonals of parallelogram bisect each other. If a pair of opposite sides of a quadrilateral are parallel and equal, the quadrilateral is a parallelogram. Ø The line–segment joining midpoints of two sides of a triangle is parallel to the third side and half of it. Conversely, the line drawn through the midpoint of one side of a triangle parallel to the other, passes through the midpoint of the third. Ø Conditions of similarity of two triangles are SAS, SSS, ASA and AA. Ø Areas of similar triangles are in the ratio of squares of corresponding sides, altitudes or medians. Ø The diagonals of a rhombus bisect each other at right angles. The figure formed by joining the midpoints of the sides of a rhombus is a rectangle. Ø The diagonals of a rectangle are equal and bisect each other. Ø The diagonals of a square are equal and bisect each other at right angles.
1 . A quadrilateral has ..... vertices, no three of which are ..... 2 . A quadri lateral is ....., if for any sid e of the quadrilateral, the line containing it has the remaining .. .. . 3 . The diagonals of a rectangle are ..... 4 . The diagonals of a parallelogram ..... each other. 5 . The line segment joining any two points lying on two ..... sides of a quadrilateral, except the two ..... lies in the interior of the quadrilateral. 6 . A ..... has one pair of opposite sides parallel to each other. 7. In a parallelogram, a pair of opposite sides are ..... and ..... 8 . In a rhombus, the diagonals ..... each other ..... 9 . Each diagonal of a rhombus ..... the angles through which it passes. 10. A diagonal of a parallelogram divides it into two ..... triangles. DIRECTIONS: For each statement given below, indicate whether it is true (T) or false (F). 11. A diagonal of a quadrilateral is a line segment joining the two opposite vertices. 12. A square is a rhombus. 13. A square is a parallelogram. 14. A rectangle is a square. 15. The diagonals of a rectangle are perpendicular. through any point on the circle. Ø From any exterior point of the circle, two tangents can be drawn on to the circle. Ø The lengths of two tangents segment from the exterior point to the circle, are equal. Ø The tangent at any point of a circle and the radius through the point are perpendicular to each other. Ø When two circles touch each other, their centres & the point of contact are collinear. Ø If two circles touch externally, distance between centres = sum of radii. Ø If two circles touch internally, distance between centres = difference of radii Ø Circles with same centre and different radii are concentric circles. Ø Points lying on the same circle are called concyclic points. (12) of (20) The sum of any two sides of a triangle is greater than the third side. IC : PTtkmml09 SOLVED EXAMPLES – LECTURE # 09
The following questions will help you get a thorough hold on the basic fundamentals of the chapter and will also help you develop your concepts. The following questions shall be covered in the class by the teacher. It is also advised that a sincere student should solve each of the given questions at leas t twice to develop the required expert ise.
DIRECTIONS: For each of the following questions, please give the complete solution. 1 . Calculate the value of x. Given that AB || CD, EF || CD and CE || FG, then ÐEFG will be A ° 65
133° 79° x B G E F ° 35
C 2 . 5 . D 6 . In the given figure, find x and y. In the pentagon given below, sum of the angles marked is y x 40° 40° D C E 7. In the given figure, AB is parallel to EF, ÐABC = 65°,
ÐCDE = 50°, ÐDEF = 30°. Find ÐBCD. O F A B D B 65° 3 . Calculate the value of x. 50° C 30° A E 3x° 8x° 4 . x° Calculate the value of x. 8 . In the given figure, APD, BPE and CPF are straight lines, find the measure of (a) the angle x, if x = c + d. (b) the angle e, if b = c = d = e = f = x. (c) the values of a + b + c + d + e + f. 37° x 68° x x IC : PTtkmml09 P
(13) of (20) 9. In the figure, AD, BE and CF are the altitudes of D ABC intersecting at G. Find the sum of ÐBAC and ÐBGC. 15. In the following figure, equal sides BA and CA of a triangle ABC are produced to Q and P respectively, so that AP = AQ. Prove that PB = QC. A P Q E F A G b r B s D B C 10. Find the number of sides of a polygon, if the sum of its angles is three times that of an octagon. C 16. ABC is an isosceles triangle such that AB = AC. Side BA is produced to a point D such that BA = AD. What is the measure of ÐBCD? D b 11. Each interior angle of a regular polygon exceeds the exterior angle by 150°. Find the number of sides of the polygon. 12. In the figure, AB, BC, CD are three sides of a regular polygon and ÐBAC is 20°. Calculate (a) the exterior angle of the polygon, (b) the number of sides of the polygon, (c)
ÐADC A B x b a b A a C c Q R a C 20° A D B 13. In the figure ABCD and ABEF are parallelograms. Prove that
DADF and DBCE are congruent. D C A a P Prove that PQR is an equilateral triangle. 18. In a trapezium ABCD, AB || DC, AB = 2CD. If the diagonals intersect at O, show that area of DAOB = 4 × area of
DC OD . 19. In the given figure, PQ || BC and ÐBPC = 90°. If PQ = 15 cm, BC = 25 cm and AP = 10.5 cm, then calculate the lengths of E F C 17. In the figure, ABC is an equilateral triangle of side 3a. P, Q and R are points on BC, CA and AB respectively such that AR = BP = CQ = a.
E B a B (a) AB and (b) PC. A 14. In the figure, if AC : CB = 1 : 3, AP = 4.5 10.5 and BQ = 7.5, PA || RC || QB, then find CR. P B B A 7.5 P R Q Q 25 C 4.5 15 C 20. P is a point on the base BC of the equilateral DABC such that BP = 1 BC. Prove that 9AP 2 = 7AB 2 . 3 A B
(14) of (20) P D C IC : PTtkmml09 Answers Success is where
preparation and
opportunity meet...
Mini Revision Tests
Answers to Mini Revision Test # 01 1. ray 2. suppl ementary. 3. transversal 4. 180° 5. 90° 6. ÐBAC or ÐCAB 7. vertical angles 9. 180° 10. parallel 8. adjacent Answers to Mini Revision Test # 02 1. True 2. True 3. False 4. True 5. False 6. 360 7. 96 3 sq.cm 8. 75% 9. 300% 10. 308 sq.cm Answers to Mini Revision Test # 03 1. 45° 2. 48° 3. 30° 4. 30° 5. 36° 6. 80° 7. 90° 8. 60° 9. 15°, 75°, 90° 10. i­iii; ii­v; iii­i, iv­ii; v­vi; vi­iv Answers to Mini Revision Test # 04 1. four, collinear 2. convex, vertices 3. equal 5. dif ferent, end­points 6. trapezium 7. equal, parallel 4. bisect 8. intersect, at 90° 9. bisects 10. congruent 11.True 12. True 13. True 14. False 15. False Solutions Solved Examples – Lecture # 09 1 . 2 . 7 . ÐECD = 30°, hence ÐEFG = 150°. \ CG || EF Required sum = 5 × 360° – Sum of angles of a pentagon Draw DH || EF, = 1800° – 540° = 1260°. \ CG || DH. We have 8x° + 3x° + x° = 180°. \ r = 50° – 30° = 20°. We have x = r = 20°. (alternate Ðs, DH || CG) 37° + x° + x° + 68° + x° = 360°. 5 . We have 133° + 90° + 79° + x° = 360°. Þ 302° + x° = 360° Þ x = 58°. 6 . x = 40° (vertically opposite Ðs ) y = 180° – 40° – x (adjacent Ðs on straight line) 65° F G D y x 5 0° C r s A H E y = 65° (alternate Ðs, CG || AB). Þ 105° + 3x° = 360°. Þ 3x° = 255° Þ x = 85°. B s = 30° (alternate Ðs, DH || EF). \ 12x° = 180° Þ x = 15°. 4 . Draw CG || AB. 30° 3 . ÐEFG = ÐFEC, ÐFEC + ÐECD = 180°, \ ÐBCD = x + y = 20° + 65° = 85°. 8 . ( a ) x = ÐCPD (vertically opposite Ðs). c + d + ÐCPD = 180° (Ðsum of D). \ c + d + x = 180°. \ x + x = 180° (given x = c + d). \ 2x = 180° Þ x = 90°. = 180° – 40° – 40° = 100°. IC : PTtkmml09 (15) of (20) ( b ) x = ÐCPD (vertically opposite Ðs). 14. Join AQ cutting CR at K. x + c + d = 180° (Ðsum of D). Þ x = 60°. CK 1 7.5 = Þ CK =
. 7.5 4 4 \ e = f = 60° (given e = f = x). (c ) a + b + c + d + e + f 9 . F 360 I = 540° – 180° = 360°. G H 2 J K Þ Consider DBGC \ ÐBGC = 180° – r – s ....(1) 7.5 13.5 21 1 +
= = 5 . 4 4 4 4 Hence PB = QC (corresponding sides of congruent triangles). ....(2) D b = 270° – 90° = 180°. A 16. Let n be the number of sides of the polygon. a a b C (n – 2) 180° = [(8 – 2)´ 180°] ´ 3. B (n – 2) 180° =6 ´ 180° ´ 3. If AB = AC, then ÐABC = ÐACB = a°(say). n – 2 = 18 Þ n = 20. Also if AC = AD, then ÐACD = ÐADC = b°(say). Let n be the number of sides of the polygon. Now in DBCD, a + a + b + b = 180° Each exterior angle = Þ a + b = ÐBCD = 180/2 = 90°. 360 o . n 17. c n - 2 h ´ 180 Each interior angle =
n In Ds ARQ, BPR and CQP, AR = BP = CQ = a (given). o
Also ÐRAQ = ÐPBR = ÐQCP = 60° (given) . AQ = BR = CP = 3a – a = 2a. \ DARQ @ DBPR @ DCQP (S.A.S.). i.e. DPQR is equilateral. As per the given condition c h D n - 2 ´ 180 ° 360 °
+ 150 =
. n n A 30n = 720 Þ n = 24. ( a ) c = 20° (base Ðs, isosceles D). ( b ) B 2x ( a ) In triangles APQ and DABC i.e. x = 40°. (i ) ÐA = ÐA (common) Let n be the number of sides. (i i ) ÐAPQ = ÐABC (corresponding Ðs, PQ || BC) (i ii ) ÐAQP = ÐACB (corresponding Ðs, PQ || BC) c n - 2 h ´ 180 ° = 140 ° n 19. or 180n – 360 = 140n. \ DAPQ ~ DABC (A.A.A.) 40n = 360 Þ n = 9. (c ) C Ds AOB and DCOD are similar. The ratio of the proportional sides is 1 : 2. Hence the ratio of the areas of the triangles COD and AOB is 1 : 4. b = 180° – 20° – 20° (Ðsum of D) = 140°. \ the exterior angle = 180° – 140° (adjacent angles on line). x O 18. 360 + 150n = 180n – 360. \ Produce AB and DC to meet at E. ÐEBC = ÐECB = 180° – 140° = 40°. ( b ) \ EB = EC (sides opposite equal angles). EB + BA = EC + CD \ EA = ED. AP PQ 10.5 15
=
=
Þ
Þ AB = 17.5 cm. AB BC AB 25 BP = AB – AP = 17.5 – 10.5 = 7 cm. In DBPC, PC 2 = BC 2 – BP 2 (Pythagoras’ Theorem) \ PC 2 = 25 2 – 7 2 = 576 Þ PC = 24 cm. ÐBEC = 180° – 40° – 40° = 100°. 20. Draw AD ^ BC. \ ÐEDA = ÐEAD (base angles, isosceles triangle). AB = BC = CA (given). BP = \ ÐEAD =
13. Q R ÐBAE + b + 90° = 180°. ÐBGC + ÐBAC = 270° – (b + r + s). 12. P In DPAB and DQAC since, PA = QA, BA = CA and ÐQAC = ÐPAB (vertically opposite angles) \ Ds PAB and QAC are congruent. Adding equation (1) and (2), 11. 7.5 K Consider DBAE. Þ ÐBAE = 90° – b 15. 3 C 1 4.5 KR 3 13.5
= Þ KR = 4.5 4 4 \ CR = CK + KR = ÐBGC + r + s = 180°. 10. A Consider D KRQ and APQ. o = 3 ´ 180° – B Consider Ds ACK and ABQ, Þ 3x = 180° ( Q c = d = x). 180 o - 100 o 2 = 40° i.e. ÐADC = 40°. AB || DC || EF (opposite sides of a ||gram). AB = DC = EF (opposite sides of a ||gram). A 1 BC (given). 3 1 BC (^ bisector theorem). 2 Also, PD = BD – BP. BD = DC = \ DF = CE (opposite sides of a ||gram). In DADF and BCE, Also, AP 2 = AD 2 + PD 2 (Pythagoras’ Theorem). ( a ) AD = BC (opposite sides of a ||gram). ( b ) AF = BE (opposite sides of a ||gram). (c ) DF = CE (proved). \ DADF @ DBCE (S.S.S.). (16) of (20) = AB 2 -
C B 1 1 1 1 = BC - BC = BC = AB . 2 3 6 6 \ CDFE is a ||gram (opposite sides are ||). P D AB 2 AB 2 7 2 +
= AB 4 36 9 \ 9AP 2 = 7AB 2 .
IC : PTtkmml09