2D Geometry Questions and Answers

PROBLEM SOLVING 1 In each diagram shown below solve for the length of side CD using the information given Show the trig ratio you use and your algebra to find your final answer Round all answers to the nearest tenth b c a D 15 E E D 75 32 43 C d D 56 35 D 27 28 B B

4 In rectangle ABCD shown below CD 24 and m ABD 35 Determine the length of diagonal BD to the nearest tenth Show the work that leads to your answer A B 35

cos u sin 2u cos 2u tan 2u 336 125 73 24 25 25 336 527 2 U X X

Students often struggle with the following statement All squares are rectangles but not all rectangles are squares Based on your knowledge of rectangles from this lesson and your knowledge of squares from middl school why is this statement true

A square has its length and width increased by two feet and four feet respectively so that it is now a rectangle The perimeter of the new rectangle is eight feet less than twice the original perimeter What was the original perimeter Show all work that leads to your answer Hint Let x square If you cannot do an algebraic approach at least attempt it by guess and check the side length of the original

C A rhombus has a perimeter of 80 inches Its longer diagonal is 32 inches Explain why the shorter diagonal must be 24 inches

In the following diagram of AQRS QR 8 and QS 17 Determine the value of sin Q Show how you arrived at your answer Q 17 8

Construct a line AB such that it is parallel to RS shown below Use only a straightedge and compass Leave all construction marks Hint plot a third point not on the line and rotate about this point S

I Given the right triangle shown below with m A 67 measure each side length to the nearest millimeter and then fill out the table For the ratios express your answers in decimal form to the nearest hundredth BC AB AC b cos cos 67 BC AC AB AC BC Making sure your calculator is in DEGREE MODE evaluate each of the following and verify that they either match or are close to the ratios you found in 1 a sin 67 c tan 67 AB A 67 C

Joey wishes to contribute three items to a gift box He aims to pick from seven particular items Find the number of ways he can select the three items from these seven items Select one a 45 b 210 c 35 d 220

In trapezoid MNPQ shown below MN OP m M 108 m P 4x 15 and m Q 3x 21 Explain why m N must be 127 M N

7 Quadrilateral EFGH has vertices at E 1 8 F 6 1 G 4 4 and H 9 5 a Prove that EFGH is a parallelogram b Prove that EFGH is not a rhombus Many methods

Given that CDEF is a trapezoid with CD FE list all pairs of supplementary angles C

The faces of a normal die are numbered from one to six Three such dice are used in an experiment The first one is thrown and the number shown on the top face is a five The second and third dice are then thrown Calculate as a fraction the probability that All three numbers shown are odd Select one a 2 3 b 1 2 Oc 1 6 d 1 4

Complete the equations which represent the relationship between polar and rectangular coordinates r sin tan x y Choose Choose Choose r 2 xr V 4

6 Quadrilateral ABCD has vertices at A 0 6 B 4 1 C 4 0 and D 8 7 Prove that a ABCD is a rhombus using the distance formula b The diagonals of ABCD are perpendicular a

A sailor is sailing around three islands She travels 2 6 km from Island A to Island B She then turns 55 and travels 2 9 km to island C How long is the last leg of the trip from Island C to Island A Select one a 1 3 km b 2 6 km OC 8 5 km d 2 9 km

A boy who is 5 feet 6 inches tall casts a shadow that is 8 feet and 3 inches long What is the measure of the angle of elevation of the sun A as shown in the picture Round to the nearest degree Be careful with your units in this problem

Consider the angle shown below whose vertex is at 2 2 5 A circle with a radius 1 7 units long is centered at the vertex of the angle and the terminal point at 3 4 3 46 9 Preview 4 3 4 3 46 F 2 2 5 3 2 Y a The terminal point is how many units to the right of the center of the circle units Preview b The terminal point is how many radit to the right of the center of the circle radi Preview c Therefore the radian measure of the angle is 3 5 I

gra Hint Graph the vectors on graph paper then draw the resultant vector Select one a b Vectors choose the image which best represents the difference of u v

Given the information a vector has length 8 with South direction it would be drawn from the the point 0 6 the origin anypoint and extend

6 x 1 y 3 4 y 6 5 4 3 2 1 6 5 4 3 2 1 11 2 31 0 45 6 6 5 4 3 2 1 4 7 x 2 y 1 16 y 6 5 4 3 2 1 6 5 4 3 21 1 0 11 2 3 4 5 6 8 x 1 y 4 9 y 6 5 4 3 2 1 0 N F1 2 6 5 4 3 2 1 31 4 5 6 6 5 4 3 2 1 1 2 3 4 5 6 9 x y 3 25 ya 0 1 2 3 4 5 6 1 1 2 3 4 5 6 X A 1 2 3 4 5 6 X

Find the value of the variable 24 A X D B Answer T 40 13 Find the value of the variable S C

Which is the equation of a circle with center 2 3 and radius r 5 Select one O a x 2 2 y 3 25 b x 2 2 y 3 25 c x 2 y 3 10 d x 2 2 y 3 10 Which is the equation of a circle with diameter AB where A 5 4 and B 1 4 Select one a x 5 2 y 4 10 b x 5 y 4 100 c x 2 2 y2 25 d x 2 y 5

the figure below what is m C D B elect one a 50 b 15 c 35 30 d 65 N d 7 5 the figure below what is the value of z 4 elect one a 2 9 Ob 8 75 c 5 6 E 100

he center and radius of a circle with equation x 3 2 y 6 4 one center 6 3 r 2 center 3 6 r 4 center 3 6 r 4 center 3 6 r 2 ck is the equation of a circle with center 3 5 that passes though the point 4 10 tone x 3 2 y 5 2 74 x 4 y 10 74 x 3 2 y 5 226 x 4 y 10 226

2 The equation x 6x y 6y 9 0 is an equation of a circle in expanded form Complete the square to find the center and radius and then write the equation of the circle in standard form To earn full credit you must show your work a center b radius c Equation

e the standard equation for a circle er 10 6 r 6 ct one a x 6 y 10 36 b x 10 y 6 6 c x 10 2 y 6 6 d x 10 2 y 6 36 heck ite the standard equation for a circle ter 6 8 that passes through 0 0 lect one a x 6 2 y 8 196 b x 6 2 y 8 14 c x 6 2 y 8 100 Od x 6 2 y 8 10

y x 4 Graph y x 4 Choose the correct graph below G OB OA 2 2 OC 2 2 OE 3 5 OG The graphs do not intersect oo Determine any points of intersection Choose the correct answer below O C OB 5 3 O D 2 2 and 2 2 OF 5 3 and 3 5 D

Write an equation of a circle with diameter AB A 6 6 B 14 14 Choose the correct answer below OA x 10 y 10 6 OC x 10 y 10 32 O E x 10 y 10 32 OG x 14 y 14 6 OB x 10 y 10 14 OD x 6 y 6 14 OF x 10 y 10 32 OH x 10 y 10 32

a building The height of the ros by h t 8t 2 80t 25

e line parallel to g x 6x 8 and passing through the point 4 2 Write the form

9 O Answer Check Find the diameter of OO A line that appears to be tangent is tangent If your answer is not a whole number round to the nearest tenth 12 to the nearest tenth 24

S hat is mST nswer 100 hat is mZQPT swer A

You focus your camera on a circular fountain Your camera is at the vertex of the angle formed by tangents to the fountain You estimate that the angle is 46 What is the measure of the arc of the circular basin of the fountain that will be in the photograph The measure of the arc of the circular basin of the fountain that will be in the photograph is Simplify your answer HETO AME

Find the value of the variable B

ngle Measures a C Question 6 12 4 17 Find the value of c using the given chord and secant lengths in the diagram shown to the right Round to the nearest tenth as needed Points 0 of 1 10 13

K What is the value of y y Type an integer or a decimal

efer to OC for problems 1 3 Segment DE is tangent to OC D C F elect one DE 4 and CE 8 what is the radius E a 8 6 b 4 3 c 4 d 2 10 DE 8 and EF 4 what is the radius

A piece of cardboard is 2 9 times as long as it is wide It is to be made into a box with an open top by cutting 2 inch squares from each corner and folding up the sides Let x represent the width in inches of the original piece of cardboard Answer the following questions CIC K 2 2 2 2 a Represent the length of the original piece of cardboard in terms of x Length 2 9x in Use integers or decimals for any numbers in the pression b Give the restrictions on x What will be the dimensions of the bottom rectangular base of the box

10 i I 1 I I T 1 I 1 I I 1 1 i 1 I 1 I I T i I L I I T I I 1 1 I T I I 1 I 1 8 Find an 6 1 I I I I 1 I I I I 1 I I I I 1 I I I I 1 T I I 1 I I I I 1 I 1 4 2 14 p u I 1 12 1 1 18 1 10 1 16 1 1 1 i 1 1 i 1 4 1 1 6 I 1 2 i I 7 10 12 714 equation for the graph above D I 1 1 1 1 T 1 1 1 1 I 1 1 1 I I 1 1 T 1 I I T I I i 1 1 I 1 1 1 1 4 6 I 1 1 I I 1 1 1 I I I 1 1 I I I I 1 I 1 T i I 1 i 1 T I I 1 i 1 T I I 1 I I T 8 10

1 A student inscribes quadrilateral ABCD inside a circle The measures of angles A B and C are given below Find the measure of each angle of quadrilateral ABCD m A 8x 4 m B 5x 4 m C 7x 4 2 A student inscribes an angle inside a semicircle to form a triangle The measures of the angles that are not the vertex of the inscribed angle are x and 2x 9 Find the measures of all three angles of the triangle Explain how you got your answer

84 a Answer Check nput answer as a b c bo a 78 Co 42 Find the value of the variables For the circle the dot represents the center b 87 36 nput answer as a b c

Find the value of x to the nearest tenth X 4 2 10

23 m 4 5x m 5 4 x 5 24 m 2 4x 3 m27 5 x 3 25 m 3 1 26 m 1 m25 x 52 II 35 3 5 6 7 8 4 N Exercises 23 30

D Exercises 17 18 18 Given Find AB DC DB bisects LADC m 1 36 m A

1 m A 63 and m B 42 2 m B 39 and m C 820 3 mA m C 57 4 m B 42 and m A m C A Exercises 1 6 03 B

In Exercises 13 to 15 j k and AABC 13 Given m3 L 1 50 m 4 72 m 1 m2 and m25 Find m B A 5 6 3 4 Exercises 13 15 2 1 C

In Exercises 9 and 10 classify the triangle not shown by considering the measures of its angles 9 a All angles of ABC measure 60 b In ADEF mLD 40 and m E 50 10 a In AXYZ m X 123 b In ARST m R 45 m S 65 and m T 70

8 a In AXYZ XY YZ b In ARST RS 6 ST 7 and RT 1 8