Math 121!
Geometry: Shapes! Prerequisite: None
Course Description
In this micro-course, we’ll work through practice problems involving rectangles (including shifting dimensions), triangles (including angle measures, Pythagorean Theorem, and special right triangles), circles (including sector area and arc length), polygons, areas of shaded regions, and figures inscribed in other figures.!
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Triangles, Triangles, Triangles!
IF YOU’RE TRYING TO FIND…!
WHAT DO YOU DO?!
the measure of an angle!
the length of a diagonal line!
the area!
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Triangles, Triangles, Triangles! WHEN YOU SEE A…! 30-degree angle or 60-degree angle!
WHAT ARE THE SIDE LENGTHS?!
60° 30°
45°
45-degree angle!
45° WHEN YOU SEE A…!
WHAT CAN YOU CREATE? !!
120-degree angle!
120° Math 121
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Circle Properties What is the radius?! ! ! !
Area of Circle = 16 !
What is the diameter?! ! ! ! What is the circumference?! ! ! ! What is the maximum distance between 2 points on the circle?! ! ! !
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Circle Properties
Central A ng le Arc = 360° Circumference Central A ng le Area o f S ector = 360° Area o f C ircle
center
central angle sector arc
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Review Questions D
B
A
xo
yo C
Note: Figure not drawn to scale.
If x = 20, and AB = BC = CD then what is the value of y? (A) 60o (B) 76o (C) 85o (D) 90o (E) 100o
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Review Questions If the value of a circle’s area is the same as the value of that circle’s circumference, what is the radius of the circle? 1 2 (B) 1 (A)
(C) 2 (D) 2 (E) 4
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Review Questions B A
C
E
D
In the polygon above, triangle ABC is an isosceles right triangle, and the triangle has an area of 72 square inches. What is the area of square ACDE, in inches?
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Review Questions R
Q
S
T
In the figure above, a square is circumscribed around a circle with a diameter of 20 cm. Points Q, R, S, and T are the midpoints of the square’s sides. What is the total area, in cm², of the shaded regions?
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Review Questions B
A
30° 2
C
In the diagram above, what is the area of ΔABC? 3 2 (B) 1 (A)
(C) 3 (D) 2 (E) 2 3
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Review Questions The length of a rectangle is 3 times its width. The area of the rectangle is 48 square inches. What is the width, in inches, of the rectangle? (A) (B) (C) (D) (E)
4 8 12 16 22
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Review Questions A square is inscribed in a circle. If the area of the inscribed square is 50, what is the area of the circle? (A) (B) (C) (D) (E)
5π 5π 2 10π 25π 50π
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Review Questions B
C 70°
A
Note: Figure not drawn to scale.
In triangle ABC above, the measure of angle A is half the measure of angle B. What is the measure of angle C? (A) (B) (C) (D) (E)
65° 70° 75° 80° 85°
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Review Questions If circle A has a circumference of 60 , and circle B has a circumference equal to the length of a 72° arc of circle A, what is the area of circle B? (A) (B) (C) (D) (E)
24 36 49 64 100
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Review Questions
In the figure above, the perimeter of ∆OAB is 25, and the circumference of the circle center at O is 12 . What is the perimeter of the shaded region? (A) (B) (C) (D) (E)
13 + 3 13 + 4 19 + 3 19 – 4 25 – 12
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Review Questions 6 6 3 10 Note: Figure not drawn to scale.
What is the perimeter of the figure shown above?
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Review Questions M x°
(100 – a)°
100° O L N Note: Figure not drawn to scale.
In the figure above, side LN of triangle LMN is on line O. What is x in terms of a ? (A) 20 – a (B) a (C) 20 + a (D) 100 – a (E) 100 – 2a
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Review Questions
x+
x
y 2
y
In the right triangle above, if y = 12, what is the value of x? (A) (B) (C) (D) (E)
5 8 9 13 15
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Review Questions The lengths of two opposing sides of a square are decreased by forty percent. By approximately what percent would the lengths of the other two sides have to be increased so that the area of the new figure (a rectangle) is the same as the area of the original square?
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Review Questions
The diagram above shows four identical circles with a total circumference of 16. Assuming that each circle is tangent to two of the other circles in the diagram, what is the circumference of the smallest circle that completely contains all four circles? (A) 4π (B) 4π 2 (C) 8π
( ) (E) 8π (1+ 2 ) (D) 4π 1+ 2
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Review Questions G F
H C K
J
Pentagon FGHJK, shown above, has equal sides and equal angles. If C is the center of the pentagon, what is the degree measure of ∠KCJ (not drawn)? (A) 60° (B) 68° (C) 70° (D) 72° (E) 75°
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Review Questions P
Q
R
In the figure above, each of the four squares has sides of length 2x. If triangle PQR is formed by joining the centers of three of the squares, what is the perimeter of triangle PQR in terms of x? (A) (B) (C) (D) (E)
4x 2 2x 2 4x + 2 4x + 2x 2 4 + 2x 2
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Review Questions B z°
y° A
x° E
C w° 55° D
Note: Figure not drawn to scale.
In ∆ABD above, E lies on segment AD, and C lies on segment BD. The measure of angle D is 55º. What is w + x + y + z ?
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Review Questions A triangle has a base with length 13, with the other two sides equal in length. If the lengths of the sides of the triangle are integers, what is the shortest possible length of a side? (A) 5 (B) 6 (C) 7 (D) 8 (E) 9
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Review Questions 1 2
1
2 1
3 I.
1
2
(A) (B) (C) (D) (E)
None I only II only III only I and II
1
1
Note: Figure not drawn to scale. All angles of the shapes above are right angles, all short sides have length 1, and all long sides have length 2. Which of the following figures could be made by using all three shapes once each without overlapping or cutting them?
1 2
II.
1 2 1
2
4
1 2 3
III.
2
4
2
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