8. if $ \triangle p q r \sim \triangle s t u $ and $ p q=8 \mathrm{~cm}, q r=12 \mathrm{~cm} $, and $ s t=4 \mathrm{~cm} $, what is the length of $ t u $?
- a. $ 11 \mathrm{~cm} $
- b. $ 6 \mathrm{~cm} $
- c. $ 4 \mathrm{~cm} $
- d. $ 8 \mathrm{~cm} $
9. what is the point called where the perpendicular bisectors of the sides of a triangle meet?
- a. incenter
- b. centroid
- c. orthocenter
- d. circumcenter
10. which angle is crucial in the sas criterion?
- a. the included angle
- b. the largest angle
- c. any angle in the triangle
- d. the smallest angle
11. a triangles angle bisectors intersect at a point equidistant from all three sides. this point is called the:
- a. centroid
- b. incenter
- c. orthocenter
- d. circumcenter
12. what is the scale factor if a triangle with sides 4,5, and 6 is similar to a triangle with sides 8,10, and 12?
- a. 4
- b. 1
- c. 2
- d. 3
13. what is the purpose of extending rays when using the asa method?
- a. to find the intersection point of the angles
- b. to verify the angle measurements
- c. to measure the longest side
- d. to label the base of the triangle
14. in aa similarity, how many pairs of corresponding angles need to be congruent?
- a. three
- b. four
- c. two
- d. one

Question

Accepted Answer

# 8.
## Step1: Use the property of similar triangles
For similar triangles $\triangle PQR\sim\triangle STU$, the ratios of corresponding sides are equal. That is $\frac{PQ}{ST}=\frac{QR}{TU}$

## Step2: Substitute the given values
Given $PQ = 8\mathrm{cm}$, $QR=12\mathrm{cm}$, $ST = 4\mathrm{cm}$. Substitute into $\frac{PQ}{ST}=\frac{QR}{TU}$, we get $\frac{8}{4}=\frac{12}{TU}$

## Step3: Solve for $TU$
Cross - multiply: $8\times TU=4\times12$, $8TU = 48$, then $TU=\frac{48}{8}=6\mathrm{cm}$

# 9.
# Brief Explanations:
The circumcenter is the point where the perpendicular bisectors of the sides of a triangle meet. The in - center is the point of intersection of angle bisectors, the centroid is the intersection of medians, and the ortho - center is the intersection of altitudes.

# Answer:
d. Circumcenter

# 10.
# Brief Explanations:
In the SAS (Side - Angle - Side) criterion for triangle congruence, the included angle (the angle between the two sides) is crucial. If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.

# Answer:
a. The included angle

# 11.
# Brief Explanations:
The in - center of a triangle is the point where the angle bisectors intersect. It is equidistant from all three sides of the triangle. The centroid is the intersection of medians (equidistant from vertices in a mass - point sense), the ortho - center is the intersection of altitudes, and the circumcenter is the intersection of perpendicular bisectors (equidistant from vertices).

# Answer:
b. Incenter

# 12.
## Step1: Calculate the scale factor
The scale factor $k$ of two similar triangles is given by the ratio of corresponding sides. Let the first triangle have sides $a = 4$, $b = 5$, $c = 6$ and the second triangle have sides $A=8$, $B = 10$, $C = 12$. Then $k=\frac{A}{a}=\frac{8}{4} = 2$, $\frac{B}{b}=\frac{10}{5}=2$, $\frac{C}{c}=\frac{12}{6}=2$

# Answer:
c. 2

# 13.
# Brief Explanations:
When using the ASA (Angle - Side - Angle) method, extending rays is used to find the intersection point of the angles. This helps in constructing or visualizing the triangle based on the given angles and side.

# Answer:
a. To find the intersection point of the angles

# 14.
# Brief Explanations:
In AA (Angle - Angle) similarity, if two pairs of corresponding angles of two triangles are congruent, then the triangles are similar. By the triangle angle - sum property (the sum of angles in a triangle is $180^{\circ}$), if two angles are congruent, the third angle is also congruent.

# Answer:
c. Two

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QUESTION IMAGE

Question

Explanation:

8.

Step1: Use the property of similar triangles

Step2: Substitute the given values

Step3: Solve for \(TU\)

9.

Answer:

10.