Download Explanations ( Geometry )

Explanations ( Geometry )
1. For any two points, (x1, y1) and (x2, y2), the midpoint is given by the following formula.
First, find the coordinates for point G and point H.
Point G is located at (-1, 4), and point H is located at (-1, -2).
Next, find the midpoint using the formula.
Therefore, the midpoint between point G and point H is (-1, 1).
2. Using examples to notice patterns and make conjectures is a type of reasoning known
as inductive reasoning.
Since the statement was based on a table and not proven for all possible values of F, V,
and E, the mathematician used inductive reasoning.
3. First find the inverse of each of the statements. Remember, the only time an "if p, then
q" statement is false is when p is true and q is false.
The inverse of a conditional statement is formed by negating both the hypothesis and
the conclusion of the conditional statement.
Statement: If an object is a cube, then it has six square faces.
Inverse:
If an object is not a cube, then it does not have six square faces. true
Statement: If an object is a cube with side length s, then its volume is calculated by s 3.
Inverse:
If an object is not a cube with side length s, then its volume is not
calculated by s3. true
Statement: If an object is a cube with side length s, then its surface area is calculated
by 6s2.
Inverse:
If an object is not a cube with side length s, then its surface area is not
calculated by 6s2. true
Statement: If an object is a cube, then it has eight vertices.
Inverse:
If an object is not a cube, then it does not have eight vertices. false
(counterexample: rectangular prism)
4. The sum of the interior angles, I, in a polygon with n sides is given by the formula
below.
I = (n - 2) × 180°
Substitute n = 6 into the formula, and solve for I.
I = (6 - 2) × 180°
= 4 × 180°
= 720°
5. The Exterior Angle Inequality states than an exterior angle of a triangle is greater than
either of the remote interior angles.
Angle KMN is an exterior angle. Therefore, it must be the largest angle.
Since the triangle is a right triangle, the m
Therefore,
K>
L must be less than 90°.
L.
So, the angles in order from greatest to least are
KMN,
K,
L.
6. The Angle-Angle-Side rule states that if two angles and a non-included side of a triangle
are congruent to two angles and the corresponding non-included side of another
triangle, then the two triangles are congruent.
It is given that H
K. Since the two triangles form a pair of vertical angles at point
J, these triangles have two adjacent angles that can be used to prove congruency. To
use the AAS rule, one of the pairs of sides that is not inclusive in the adjacent angles
must be proven congruent.
Sides GH
LK or sides GJ
LJ could be used to prove congruency by the AAS rule.
So, the answer is III or IV.
7. A square, a rectangle, and a rhombus all have 2 sets of parallel sides. A trapezoid only
has one set of parallel sides.
Therefore, the quadrilateral is best described as a trapezoid.
8. A median of a triangle is a line joining a vertex with the midpoint of the opposite side.
9. Let
be the angle of elevation in the right triangle. Therefore, the following is true.
The question gives that the side opposite
Label this variable
.
It also gives that the side adjacent to
represents the height of the telephone pole.
is equal to 20 ft and
= 57°.
Therefore, the following is true.
Now, solve for
.
The telephone pole is about 31 feet tall.
10 Use the given arc measurement to solve for the measure of
.
m R = 1/2(m
-m
)
=
1
=
1
R.
/2[(360° - 136°) - 136°]
/2(224° - 136°)
= 44°
11 When the diameter of a sphere is increased or decreased, the volume of the new sphere
.
is the volume of the old sphere multiplied by the cube of the scale factor.
In this problem, the diameter increased from 6 units to 12 units.
The scale factor from the original sphere to the larger sphere is
/6, or 2.
12
Therefore, the volume of the larger sphere is the volume of the original sphere
multiplied by 23, or 8.
12 The equation of a circle with center (h,k) and radius r in standard form is (x - h)2 + (y .
k)2 = r2.
It is given in the question that h = 1 and k = 9; however, r is not given.
In order to find r, calculate the distance between the center of the circle and the point
on the circle.
Since the center and the given point both have the same y-coordinate, the distance
between the two points can be found by looking at the difference in the x-coordinates.
|2 - 1| = 1 = radius
Using these values in the standard form given above will give the correct equation of the
circle described in the question.
(x - h)2 + (y - k)2 = r2
(x - 1)2 + (y - 9)2 = 12
(x - 1)2 + (y - 9)2 = 1
13 Both points lie on the horizontal line y = 19.
.
Since the points lie on a horizontal line, the distance between the two points is the
absolute value of the difference of the x-coordinates.
|-3 - 12| = 15
Therefore, the distance between the two points is 15 units.
14 For any two points (x1, y1), (x2, y2), the midpoint is given by
.
x1 + x2
y1 + y2
(
,
)
2
2
If the midpoint between (3, 4) and (x2, y2) is (6, 3), then
(6, 3) = (
3 + x2
2
,
4 + y2
)
2
(12, 6) = (3 + x2, 4 + y2)
(9, 2) = (x2, y2)
15 Only sides where shown to be congruent in the proof.
.
Therefore, reason 4 should be SSS congruence.
16 The converse of a conditional statement is formed by interchanging the hypothesis and
.
the conclusion of the conditional statement.
The converse of the given conditional statement is shown below.
If an angle is obtuse, then it is greater than 90°.
17 The sum of the interior angles I in a polygon with n sides is given by
.
I = (n-2) × 180°
It is given that I = 1,440°. Solve for n.
1,440° = (n - 2) × 180°
8=n-2
10 = n
Thus, the polygon must have 10 sides.
18 The Side-Angle Inequality says that if one angle of a triangle has a greater measure
.
than another angle, then the side opposite of the greater angle is longer than the side
opposite the lesser angle.
It is given that m
P = 90° and m
Calculate the degree of
Q.
R = 65°.
m
Q = 180°-90°-65°
m Q = 25°
The angles from smallest to largest are Q, R,
P.
Invoking the theorem, the sides least to greatest will be RP, PQ, QR.
19 The HL theorem states that if the hypotenuse and a leg of one right triangle are
.
congruent to the hypotenuse and corresponding leg of another right triangle, then the
two triangles are congruent.
There is not enough information to determine congruence in figure X, figure Y uses the
SAS theorem or the ASA theorem, and figure Z uses the ASA theorem.
Figure W shows that the hypotenuse and a leg of one triangle are congruent to the
hypotenuse and corresponding leg of the second triangle, therefore using the HL
theorem to show that RST
UVW.
20 In a parallelogram, opposites angles of a parallelogram are congruent. So, 1
.
and 2 are congruent. Similarly, 2 and 3 are congruent. By the Transitive Property
of Congruence, 1 is congruent to 3.
21 The three angle bisectors of a triangle meet in one point called the incenter.
.
The incenter is also the center of the incircle, the circle inscribed in the triangle.
22 The length of the diagonal of the illuminated area will be the hypotenuse of the triangle
.
formed.
Since the angle at the ground and the diagonal are known, the cosine equation should
be used.
23 A secant line is a line which passes through at least two points of a curve. Examine the
.
picture and note that there are two secant lines which pass through this circle.
The angle created by two secant lines has a measure that is equal to half of the
difference between the corresponding arc measurements. For this circle, this comes to
m BCD = 1/2(m
-m
)
So, use the given arc measurement and the angle measurement to solve for m
m BCD = 1/2(m
23° = 1/2(m
46° = m
84° = m
-m
- 38°)
- 38°
)
.
24 When the radius of a sphere is increased or decreased, the volume of the new sphere is
.
the volume of the old sphere multiplied by the cube of the scale factor.
In this problem, the radius decreased from 21 units to 7 units.
The scale factor from the original sphere to the smaller sphere is 7/21, or 1/3.
Therefore, the volume of the smaller sphere is the volume of the original sphere
multiplied by (1/3)3, or 1/27.
25 The equation of a circle with center (h,k) and radius r in standard form is (x - h)2 + (y .
k)2 = r2.
It is given in the question that h = 4 and k = 5; however, r is not given.
In order to find r, calculate the distance between the center of the circle and the point
on the circle.
Since the center and the given point both have the same x-coordinate, the distance
between the two points can be found by looking at the difference in the y-coordinates.
|14 - 5| = 9 = radius
Using these values in the standard form given above will give the correct equation of the
circle described in the question.
(x - h)2 + (y - k)2 = r2
(x - 4)2 + (y - 5)2 = 92
(x - 4)2 + (y - 5)2 = 81
26 If the given ordered pairs are graphed on a coordinate plane, they will form the
.
hypotenuse of a right triangle. Use the lengths of the legs of the right triangle to find
the length of the hypotenuse.
Subtract the x-coordinates to find the length of one leg.
5 - (-3) = 8
Subtract the y-coordinates to find the length of the other leg.
9-4=5
Now, the Pythagorean theorem can be used to find the length of the hypotenuse, which
is the distance between the two points.
a2 + b2 = c2
82 + 52 = c2
64 + 25 = c2
89 = c2
=c
Therefore, the distance between the two points is
units.
27 For any two points, (x1, y1) and (x2, y2), the midpoint is given by the following formula.
.
First, find the coordinates for point C and point D.
Point C is located at (-1, -3), and point D is located at (4, -3).
Next, find the midpoint using the formula.
Therefore, the midpoint between point C and point D is (1.5, -3).
28 A counterexample is an example that shows a conjecture to be false.
.
In this case, a 4×4 rectangle and a 1×15 rectangle would provide a counterexample
to the given conjecture. The 4×4 rectangle has an area and perimeter of 16 while the
1×15 rectangle has an area of 15 and a perimeter of 32. So, the 4×4 rectangle has a
larger area but smaller perimeter.
Therefore, the conjecture is false.
29 The converse of a conditional statement is formed by interchanging the hypothesis and
.
the conclusion of the conditional statement.
The converse of the given conditional statement is shown below.
If a quadrilateral is a rectangle,
then it has four right angles.
30 A hexagon is a polygon with six sides and six angles. The sum of the interior angles of a
.
polygon equals 180(n - 2), where n is the number of sides.
180(6 - 2) = 720°
Assuming all the angles in the hexagon are congruent, find the measure of one interior
angle.
720° ÷ 6 = 120°
If an interior angle equals 120°, then its corresponding exterior angle will equal 60°,
because the angles are supplementary (their sum is 180°).
60° × 6 = 360°
Therefore, the sum of the exterior angles of a hexagon is 360°.
31 The Exterior Angle Inequality states than an exterior angle of a triangle is greater than
.
either of the remote interior angles.
Since
A is a remote interior angle and
m
BCD is an exterior angle,
BCD > m
A.
32 The AAS corollary states that if two angles and a non-included side of one triangle are
.
congruent to two angles and a non-included side of another triangle, then the two
triangles are congruent.
In the drawing, James marked the included side on each triangle as congruent;
therefore, the triangles are not congruent by the AAS corollary.
33 Since it is given that GHJK is a rectangle, then by definition,
.
a. the four angles are right angles
b. the opposite sides are congruent
c. opposite sides are parallel.
Statement a satisfies reason I.
Since by definition of a rectangle, opposite sides are congruent, all that needs to be
shown for reason II is that adjacent sides are congruent.
To show that adjacent sides are congruent, use the distance formula for GH and HJ.
Therefore, GH
HJ and reason II is satisfied.
Since both pairs of sides are parallel, then reason III (only one pair of parallel sides) is
incorrect.
Therefore, the correct statements would be I and II.
34 A perpendicular bisector is a line or segment that passes through the midpoint of a
.
side of a triangle and is perpendicular to that side.
35 In a right triangle, the cosine of an acute angle is equal to the length of the adjacent
.
side divided by the length of the hypotenuse.
Use the information given in the problem to find cos(45°).
36 Notice that MAP is an inscribed angle with intercepted arc MP.
.
By the Inscribed Angle Theorem, the measure of an angle inscribed in a circle is equal to
one-half the measure of the intercepted arc.
So, m
MAP = one-half the measure of arc MP.
The measure of
MAP is 34°.
37 When the diameter of a sphere is increased or decreased, the surface area of the new
.
sphere is the surface area of the old sphere multiplied by the square of the scale factor.
In this problem, the diameter decreased from 32 units to 8 units.
The scale factor from the original sphere to the smaller sphere is 8/32, or 1/4.
Therefore, the surface area of the smaller sphere is the surface area of the original
sphere multiplied by (1/4)2, or 1/16.
38 The equation of a circle with center (h,k) and radius r in standard form is (x - h)2 + (y .
k)2 = r2.
It is given in the question that h = 6 and k = 4; however, r is not given.
In order to find r, calculate the distance between the center of the circle and the point
on the circle.
Since the center and the given point both have the same x-coordinate, the distance
between the two points can be found by looking at the difference in the y-coordinates.
|14 - 4| = 10 = radius
Using these values in the standard form given above will give the correct equation of the
circle described in the question.
(x - h)2 + (y - k)2 = r2
(x - 6)2 + (y - 4)2 = 102
(x - 6)2 + (y - 4)2 = 100
39 Both points lie on the horizontal line y = 19.
.
Since the points lie on a horizontal line, the distance between the two points is the
absolute value of the difference of the x-coordinates.
|-3 - 6| = 9
Therefore, the distance between the two points is 9 units.
40 For any two points (x1, y1), (x2, y2), the midpoint is given by
.
x1 + x2
y1 + y2
(
,
)
2
2
If the midpoint between (-3, -3) and (x2, y2) is (0, -5), then
(0, -5) = (
-3 + x2
2
,
-3 + y2
2
)
(0, -10) = (-3 + x2, -3 + y2)
(3, -7) = (x2, y2)
41 Use the distance formula and the coordinates of the points to determine the length of
.
each side of the figure.
The slopes of the sides can be determined by using the formula,
.
Since the figure has no congruent sides and the slopes are neither equal nor negative
reciprocals, the most specific name for this shape is a scalene triangle.
42 The inverse of a conditional statement is formed by negating both the hypothesis and
.
the conclusion of the conditional statement.
The inverse of the given conditional statement is shown below.
If the perimeter of a square is not 20 cm, then its area is not 25 cm 2.
43 First, add together the angle measurements that are given.
.
56° + 241° + 143° + 70° + 87° + 269° + 87° + 192°= 1,145°
Next, find the sum of the interior angles in a polygon with nine sides.
The sum of the interior angles I in a polygon with n sides is given by
I = (n-2) × 180°
When n = 9,
I = (9 - 2) × 180° = 7 × 180° = 1,260°
To find the value of x, subtract the sum of the given angles from the sum of the interior
angles of a polygon with nine sides.
1,260° - 1,145° = 115°
44 The Exterior Angle Inequality states than an exterior angle of a triangle is greater than
.
either of the remote interior angles.
Since
M is a remote interior angle and
Likewise, since
m M.
RHM is an exterior angle, m
M is a remote interior angle and
Therefore, the measures of
of M.
RHM and
RHM > m
BAM is an exterior angle, m
M.
BAM >
BAM are both greater than the measure
45 The Angle-Side-Angle rule states that if two angles and the included side of one triangle
.
are equal to two angles and the included side of another triangle, then the two triangles
are congruent.
It is given that
W
X.
Since the two triangles form a pair of vertical angles at point Y, these triangles have two
adjacent angles that can be used to prove congruency.
To use the Angle-Side-Angle rule, the included sides must be proven congruent.
Therefore, it is necessary for WY to be congruent to XY.
46 A rectangle is a parallelogram, and neither a trapezoid nor an isosceles trapezoid could
.
have exactly two opposite right angles.
So, a quadrilateral that is not a parallelogram and could have exactly two opposite right
angles must be a kite.
47 The centroid is located at the intersection of the medians of a triangle.
.
48 In the right triangle given in the problem the two legs have the same length. This
.
indicates that the trianlge is a 45°-45°-90° triangle, or isosceles right triangle.
Since
NOM is one of the congruent base angles, m
NOM equals 45°.
49 Since line TQ is tangent to circle S at point Q and chord RQ is a diameter of circle S, line
.
TQ is perpendicular to chord RQ. Thus, RQS is a right angle.
So,
RQP and
PQT are complementary angles.
m
RQP + m
PQT = 90°
27° + m
PQT = 90°
m
PQT = 90° - 27°
m
PQT = 63°
50 The volume of a sphere with radius, r, is shown below.
.
Find the volume of the sphere using the formula and the given radius.
Therefore, the approximate volume of the sphere is 1,633.39
cm3.