Download 2 - Geometry And Measurement

Geometry and Measurement
Circles, Lines , quadrilaterals and other polygons are done by:
Ali Mohammed Ali
Grade 12-04
Circles
 A circle is the locus of all points
equidistant from a central point.
Definitions Related to
Circles:
 arc: a curved line that is part of the circumference of a
circle.
 circumference: the distance around the circle.
 diameter: the longest distance from one end of a circle to
the other.
 pi : A number, 3.141592..., equal to (the circumference) /
(the diameter) of any circle.
 radius: distance from center of circle to any point on it.
 tangent of circle: a line perpendicular to the radius that
touches ONLY one point on the circle.
Related questions to
Circles.
Question #1
In a large field, a circle with an area of 144π square meters is drawn out. Starting
at the center of the circle, a groundskeeper mows in a straight line to the circle's
edge. He then turns and mows ¼ of the way around the circle before turning
again and mowing another straight line back to the center. What is the length, in
meters, of the path the groundskeeper mowed?
a)
24 + 6π
b)
12 + 6π
c)
12 + 36π
d)
24 + 36π
e)
24π
Question #2
If the area of a circle is four times larger than the
circumference of that same circle, what is the diameter of
the circle?
a) 4
b) 16
c) 32
d) 2
e) 8
Question # 3
A star is inscribed in a circle with a diameter of 30, given
the area of the star is 345, find the area of the shaded
region, rounded to one decimal.
a) 351.5
b) 356.5
c) 341.5
d) 346.5
e) 361.5
Question #4
In the figure above that includes Circle O, the measure of angle
BAC is equal to 35 degrees, the measure of angle FBD is
equal to 40, and the measure of arc AD is twice the measure of
arc AB. Which of the following is the measure of angle CEF?
The figure is not necessarily drawn to scale, and the red
numbers are used to mark the angles, not represent angle
measures.
a) 30
b) 75
c) 60
d) 110
e) 80
Question #5
A park wants to build a circular fountain with a walkway
around it. The fountain will have a radius of 40 feet, and
the walkway is to be 4 feet wide. If the walkway is to be
poured at a depth of 1.5 feet, how many cubic feet of
concrete must be mixed to make the walkway?
a) None of the other answers
b) 1296π cubic feet
c) 504π cubic feet
d) 336π cubic feet
e) 1936π cubic feet
Answers
Explanation for question #1
Circles have an area of πr2, where r is the radius. If this circle has an area
of 144π, then you can solve for the radius:

πr2 = 144π

r 2 = 144

r =12
When the groundskeeper goes from the center of the circle to the edge,
he's creating a radius, which is 12 meters. When he travels ¼ of the way
around the circle, he's traveling ¼ of the circle's circumference. A
circumference is 2πr. For this circle, that's 24π meters. One-fourth of
that is 6π meters. Finally, when he goes back to the center, he's creating
another radius, which is 12 meters. In all, that's 12 meters + 6π meters + 12
meters, for a total of 24 + 6π meters.
Answer is A
Explanation for question #2
Set the area of the circle equal to four times the
circumference πr2 = 4(2πr). Cross out both
π symbols and one r on each side leaves you with r
= 4(2) so r = 8 and therefore d = 16.
Answer is B
Explanation for question #3
The area of the circle is (30/2)2*3.14 (π) = 706.5,
since the shaded region is simply the area
difference between the circle and the star, it’s
706.5-345 = 361.5
Answer is E
Explanation for question #4
AD + AB + CD + BC = 360
AD + AB = 210
AD + AB + 80 + 70 = 360
Because AD = 2AB, we can substitute 2AB for AD.
2AB + AB = 210
3AB = 210
AB = 70
This means the measure of arc AB is 70 degrees, and the
measure of arc AD is 2(70) = 140 degrees.
Now, we have all the information we need to find the
measure of angle CEF, which is equal to half the
difference between the measure of arcs AD and CD.
CEF = (1/2)(140 - 80) = (1/2)(60) = 30.
The answer is 30.
Answer is A
We are searching for the surface area of the
shaded region. We can multiply this by the
depth (1.5 feet) to find the total volume of this
area. The radius of the outer circle is 44
feet. Therefore its area is 442π = 1936π. The
area of the inner circle is 402π =
1600π. Therefore the area of the shaded area
is 1936π – 1600π = 336π. The volume is 1.5
times this, or 504π.
Answer is C
Quadrilaterals And
Other Polygons
Quadrilateral just means "four sides". But the sides have to be
straight.
There are special types of quadrilateral:
 The Rectangle it s a four-sided shape where every angle is a right
angle (90°).
 The Rhombus it is a four-sided shape where all sides have equal
length.
 The Square it has equal sides and every angle is a right angle
(90°).
 The Parallelogram it has opposite sides parallel and equal in
length. Also opposite angles are equal.
 The Trapezoid it has a pair of opposite sides parallel.
Questions related to
Quadrilaterals And Other
Polygons
1-Which quadrilateral does not have two sets of parallel sides?
a)
Square
b)
Rhombus
c)
Trapezoid
d)
Parallelogram
2-identify the following:
a)
Rectangle
b)
Triangle
c)
Square
d)
Parallelogram
 3- Based on the knowledge you have of triangles, what
do you know about a parallogram? Measure the angles
of the parallelogram with a protractor, and add them up.
What is the sum?
 4- Based on the knowledge you have of triangles, what
do you know about a trapezoid? Measure the angles of
the trapezoid with a protractor, and add them up. What
is the sum?
Answers
 1-trapezoid
 2- parallelogram
 3- 2 opposite equal angles, parallel sides, 2 sets
equal sides angle sum is 360 degrees.
 4- 1 set parallel sides, 4 vertices, unequal
sides total angles = 360 degrees
Parallel and Intersecting Lines
When a line intersects (or crosses) a pair of parallel lines, there are
some simple rules that can be used to calculate unknown angles.
 a=b(and c=d, and e=f) these are called vertically opposite angles.
 a=c(and b=d) these are called corresponding angles.
 b=c these are called alternate angles.
 a+e=180 degree, because adjacent angles on a straight line add
up to 180 degree. these are called supplementary angles.
Note also, that c+e=180degree( allied or supplementary angles)
Questions related to
Parallel and Intersecting
Lines
1) Which line below is parallel to y-2=3/4x?
2) assume line a and line b are parallel if angle x is three
bigger than twice the square of four of angle, then what is
angle y?
3) Two line are described by the equations:
y=3x+5 and 5y-25=15x which of the following is true
about the equation for these two lines?
4) A line passes through the points(-1,-2) and (1,2). Which
of the following lines is parallel to this line?
Answers
1. y=3/4x-5
2. 7
3. They represent the same lines.
4. The line between the points (-2,0) and (0,4).
5. 4x – 3y=2 , 6y= 8x+9
Thank you
Solid Geometry
Made by: Ali Khalfan Ali
Grade: 12-04
Teacher Name: Mr. Abdul Salam Abdulla
Dulaimi
Information about:
 Solid Geometry:

In mathematics, [solid geometry] was the traditional name for the geometry of threedimensional Euclidean space — for practical purposes the kind of space we live in.

It was developed following the development of plane geometry.

Stereometry deals with the measurements of volumes of various solid figures including
cylinder, circular cone, truncated cone, sphere, and prisms.

The Pythagoreans had dealt with the regular solids, but the pyramid, prism, cone and
cylinder were not studied until the Platonists.

Eudoxus established their measurement, proving the pyramid and cone to have onethird the volume of a prism and cylinder on the same base and of the same height, and
was probably the discoverer of a proof that the volume of a sphere is proportional to the
cube of its radius.
Some videos about [Solid
Geometry]:
Some videos about [Solid
Geometry]:
Some [Solid Geometry] Shapes:
Some Question about [Solid Geometry]:
1. A cubic box has sides of length x. Another cubic box
has sides of length 4x. How many of the boxes with
length x could fit in one of the larger boxes with side
length 4x?
A. 16
B. 40
C. 4
D. 80
E. 64
Some Question about [Solid Geometry]:
2. A cube weighs 5 pounds. How much will a different cube of the
same material weigh if the sides are 3 times as long?
A. 10 pounds
B. 135 pounds
C. 45 pounds
D. 15 pounds
E. 50 pounds
Some Question about [Solid Geometry
 3. I have a hollow cube with 3” sides suspended inside a
larger cube of 9” sides. If I fill the larger cube with water and
the hollow cube remains empty yet suspended inside, what
volume of water was used to fill the larger cube?
A. 72 in 3
B. 216 in3
C. 702 in3
D. 73 in3
E. 698 in3
Some Question about [Solid
Geometry]:
4.
If the volume of a cube is 50 cubic feet, what is the
volume when the sides double in length?
A. 300 cu ft
B. 200 cu ft
C. 100 cu ft
D. 500 cu ft
E. 400 cu ft
Some Question about [Solid
Geometry]:
5. A cube is inscribed in a sphere of radius 1 such that all 8
vertices of the cube are on the surface of the sphere. What
is the length of the diagonal of the cube?
A. √(2)
B. 2
C. √(3)
D. 8
E. 1
Explain The Answers:
1. [E], because the volume of a cubic box is given by
(side length)3. Thus, the volume of the larger box is
(4x)3 = 64x3, while the volume of the smaller box is
x3. Divide the volume of the larger box by that of the
smaller box, (64x3)/(x3) = 64.
2. [B], because cube that has three times as long sides
is 3x3x3=27 times bigger than the original. Therefore,
the answer is 5x27= 135.
Explain The Answers:
 3. [C], because the volume of both cubes and then
subtract the smaller from the larger. The large cube
volume is 9” * 9” * 9” = 729 in3 and the small cube is 3”
* 3” * 3” = 27 in3. The difference is 702 in3.
 4. [E], because using S as the side length in the original
cube, the original is s*s*s. Doubling one side and
tripling the other gives 2s*2s*2s for a new volume
formula for 8s*s*s, showing that the new volume is 8x
greater than the original.
Explain The Answers:
 5. [B], because since the diagonal of the cube is a line
segment that goes through the center of the cube (and
also the circumscribed sphere), it is clear that the
diagonal of the cube is also the diameter of the
sphere. Since the radius = 1, the diameter = 2.
Thanks For Your Attention
Triangles
Done by: Omar Ali Ahmed Ali Shaheen
G12-04
What is SAT?
 The SAT and SAT Subject Tests are a suite of tools
designed to assess your academic readiness for
college. These exams provide a path to opportunities,
financial support and scholarships, in a way that's fair
to all students. The SAT and SAT Subject Tests keep
pace with what colleges are looking for today,
measuring the skills required for success in the 21st
century.
Triangles
 A triangle has three sides and three angles.
 The three angles always add to 180°
 There are three special names given to triangles
Equilateral,
Isosceles
andareScalene
that tell how many
sides (or angles)
equal. There can be 3, 2 or no equal
sides/angles:
Equilateral Triangle
 Three equal sides

 Three equal angles, always 60°
Isosceles Triangle
 Two equal sides
 Two equal angles
Scalene Triangle
 No equal sides
 No equal angles
What Type of Angle?
 Triangles can also have names that tell you what type
of angle is inside.
Acute Triangle
 All angles are less than 90°
Right Triangle
 Has a right angle (90°)
Obtuse triangle
 Has an angle more than 90°
Combining the Names
 Sometimes a triangle will have two names, for
example:
 Right Isosceles Triangle
 Has a right angle (90°), and also two equal
angles Can you guess what the equal angles are?
Perimeter
 The perimeter is the distance around the edge of
the triangle: just add up the three sides:
Area
 The area is half of the base times height.
 "b" is the distance along the base
 "h" is the height (measured at right angles to the base)
 Area = ½ × b × h The formula works for all triangles.

Another way of writing the formula is bh/2
Example

Example: What is the area of this triangle?

Height = h = 12

Base = b = 20

Area = ½ × b × h = ½ × 20 × 12 = 120
 Just be sure the "height" is measured at right
The base
angles to the "base”
Why is the Area
"Half of bh"?
 Imagine you "doubled" the triangle (flip it around one of
the upper edges) to make a square-like shape (it would
be a "parallelogram" actually), THEN the whole area
would be bh (that would be for both triangles, so just
one is ½ × bh).
For instance
 2. A right circular cylinder has a radius of 3 and a height of 5.
Which of the following dimensions of a rectangular solid will
have a volume closest to the cylinder.
Question 1

 (A) 4, 5, 5,
 (B) 4, 5, 6,
 (C) 5,5,5,
 (D) 5,5,6,
 (E) 5,6,6

V = πr2h
>
V = π × 32 × 5 = 45π
V = 45 × 3.142 = 141.39

We now have to test the volume of each of the rectangular to find out which is the closest to
141.39.

(A) 4 × 5 × 5 = 100

(B) 4 × 5 × 6 = 120

(C) 5 × 5 × 5 = 125

(D) 5 × 5 × 6 = 150

(E) 5 × 6 × 6 = 180

Answer: (D) 5, 5, 6
Solution
Question 2
 In the figure below, what is the value of y?
 (A) 40
 (B) 50
 (C) 60
 (D) 100
Solution
 Step 1: Vertical angles being equal allows us to fill in
two angles in the triangle that
y° belongs to.
 Sum of angles in a triangle = 180°. So, y° + 40° + 80°
= 180°
 y° = 60°
> y° + 120° = 180°
Vidoes
 http://www.youtube.com/watch?v=xz6gBA0M9FY
 http://www.youtube.com/watch?v=KVFwRA7kcLY
 http://www.youtube.com/watch?v=XFh_JC7OSrg
 http://www.youtube.com/watch?v=nfMkORv6ybc