NAME ______________________________
MIDTERM REVIEW- GEOMETRY
1) a. List the transformations that are rigid motions:
b. State the transformation that is not a rigid motion and explain why:
2) By constructing different types of lines in a triangle we get points of Concurrency, or a point where
three or more lines meet. Name the points of concurrency below:
A point formed by three angle bisectors in a triangle is called the: ________________________
A point formed by three perpendicular bisectors in a triangle is called the: _________________
A point formed by three medians in a triangle is called the: ______________________________
3) Angles that form a straight line add to ____________o, find the angle measures in the diagram below:
x+12
42o
4x+6
4) Sometimes it’s helpful to draw an ___________________ line to find a missing angle when you are
given a diagram like the one illustrated below. Draw the line and then solve for x:
5) The triangle DEF is formed by drawing the midsegements of triangle ABC. Find the perimeter of
quadrilateral ADFE if; BD=4, AC=10, DE=6.
6) Listed below are several notations for the transformations we have discussed. Write, being as specific
as possible, what each notation means:
Rp,-60_______________________________________________________________________________
rmn _________________________________________________________________________________
𝑇𝑣⃗ _________________________________________________________________________________
Dp,4 ________________________________________________________________________________
7) Triangles can be proven similar (~) by the following methods: AA (Angle, Angle), SAS (Side, angle, side
similarity), SSS (side, side, side similarity).
Which of the methods could be used to prove the triangles are similar? Show your work:
6
4
6
9
5
7.5
The two triangles below are NOT similar, explain why:
20o
20o
9
10
12
14
8) The median of a triangle goes from the vertex of an angle to the ______________________ of the
opposite side. Triangle ABC has medians AD and BE and CF that intersect at a point G. If FG=5, what is
the length of CG?
9) Figure A is mapped onto figure B below. Is the scale factor less than 1 or greater than 1?____________
Find the center of dilation of the two similar figures below:
B
A
10) a. Draw a right triangle ABC with right angle at B. Then draw an altitude from B to side AC and label
the point of intersection D.
b. The length of AD=20, the length of DC=5, find the length of the altitude, BD.
c. Find the length of the two legs of the right triangle AB and BC.
11) A regular octagon is drawn below. What is the least amount of degrees, in the counterclockwise
direction, would you need to turn the octagon to show rotational symmetry.
12.) In triangle ABC, AD is an angle bisector. Side AB=14, AC=10, DC=5, what is the length of BD?
B
D
A
C
13.) In the diagram below WX is parallel to YZ, fill in the proportions that could be used to find a missing
segment in the diagram:
a.
𝑉𝑊
𝑊𝑌
=
𝑋𝑍
b.
𝑉𝑊
𝑉𝑋
=
𝑊𝑌
c.
𝑉𝑊
=
𝑊𝑋
𝑌𝑍
d. If the length of VW=3, WY=12, WX=2.5, what is the length of YZ?
14.) Think about what we know about the ratio of sides, perimeters and areas. For example, if the ratio of the
sides of two triangles is 2:5, then the ratio of the perimeters is _______ and the ratio of the area is ________.
Now try this example:
The sides of a triangle are 5,6 and 8. The shortest side of a similar triangle is 15, what is the the
perimeter of the larger teriangle?
15.) Find the coordinates of A’, the image of point A(3,-1) after the composition of transformations listed
below:
rx-axis(Do,1/2(𝐴))
16.) Find the missing sides in the triangle below:
10
17.) Remember, if SinA=CosB then A+B= _________
𝜃
a. Find the value of  if Sin(4 + 20) = 𝐶𝑜𝑠(𝜃 − 10)
b. If the sinA= 3x+0.6 and CosB=4x-0.1, find the value of x.
18.) In an equilateral triangle, the length of the sides are 12. Find the length of the altitude of the triangle,
write your answer in simplest radical form.
19.) Triangle ABC can be mapped onto triangle PQR, write in function notation the transformation that would
prove that the two triangles are congruent.
20.) In the diagram of ABC below, 𝐷𝐸 ∥ 𝐵𝐶, AD=4, DB=3, and DE=10, find the length of BC.
A
D
E
B
C
21.) Draw two similar triangles, ABC MOP.
If ∡𝐴 = 42, ∡𝐵 = 56, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 ∡𝑃?
22.) In the diagram below 𝐴𝐵 ⊥ 𝐵𝐷, 𝐸𝐶 ⊥ 𝐵𝐷, ∡𝐸𝐷𝐴 = 35, find the measure of ∡𝐴𝐷𝐵.
A
E
B
C
D
23.) The two squares below have been M and N by finding the center point of the square. Square M has a side
length of 4 and square N has a side length of 12. Write a composition of transformations that would map
Square M onto Square N.
12
4
N
M
24.) The triangles below are similar.
a.) write a similarity statement:_______________
b.) one triangle can be mapped onto the other through a series of transformations, state the
transformations.
c.) What is the center of the rotation?
d.) What is the scale factor of the dilation?
25.) In right triangle ABC with right angle B, SinA=5x+.6 and CosC=3x-.4 Determine the value of x, explain your
answer.
26.) In the diagram below ∡𝐵𝐶𝐷 is an _____________________ angle,
∡𝐴 and ∡𝐵 are _______________ ___________________ angles.
Use the rule: the Exterior angle= sum of the remote interior angles to solve for x.
B
X+20
A
5x+10
2x+16
C
D
27.) Given the two sides in the right triangle below, find the third side then fill in the trig ratios:
A
18
B
C
24
SinA= ----------
SinC= ---------
CosA= ----------
CosC= ---------
TanA= ----------
TanC= ---------
28.) Given: Δ𝐴𝐵𝐶, ∡𝐴 ≅ ∡𝐶, 𝑎𝑛𝑑 𝐵𝐷 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝐴𝐶
B
Prove: Δ𝐴𝐷𝐵 𝑖𝑠 𝑎 𝑟𝑖𝑔ℎ𝑡 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒
A
C
D
29.) Using a compass and a straight edge, construct an altitude of triangle PQR. [leave all construction marks]
P
R
Q
30.) Use a compass and straight edge to reflect triangle ABC over the line drawn and label its image A’B’C’
If ∡𝐵 = 2𝑥 + 5 𝑎𝑛𝑑 ∡𝐵 ′ = 3𝑥 − 15, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒 𝐵?
If B’A’= 3√2, 𝑤ℎ𝑎𝑡 𝑖𝑠 𝑡ℎ𝑒 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑖𝑑𝑒 𝐴′ 𝐶 ′ ?
31.) In the diagram below 𝐷𝐸 ∥ 𝐹𝐺 ∥ 𝐶𝐵, 𝑚∡𝐴𝐺𝐹 = 𝑥 + 20, 𝑚∡𝐵 = 2𝑥 − 10,
a.) Find the measure of angle B
b.) Use a compass and straight edge to dilate
segment ED by a scale factor of 3 with the center of
dilation at A, label the new segment PQ.
c.) If segment ED has a length of 15, what is the length of PQ?