Question

two triangles are similar. the measure of the shortest side of the first triangle is 12 and the measure of the shortest side of the second triangle is 5. if the longest side of the first triangle measures 15, what is the measure of the longest side of the second triangle?
24
2.4
36
6.25
2 multiple choice. 4 points
if $\triangle mnp \cong \triangle vwx$ and $\overline{pm}$ is the shortest side of $\triangle mnp$, what is the shortest side of $\triangle vwx?$
$\overline{wx}$
$\overline{np}$
$\overline{xv}$
$\overline{vw}$
3 multiple choice. 4 points
the lengths of the sides of a triangle are 8, 15, and 17. if the longest side of a similar triangle is 51, what is the length of the shortest side?
16
32
24
4

Explanation:

Question 1

Step1: Recall similarity ratio

For similar triangles, the ratio of corresponding sides is equal. Let the longest side of the second triangle be \( x \). The ratio of shortest sides is \( \frac{12}{5} \), and the ratio of longest sides is \( \frac{15}{x} \). Since they are equal, we set up the proportion: \( \frac{12}{5}=\frac{15}{x} \)

Step2: Solve for \( x \)

Cross - multiply: \( 12x = 15\times5=75 \). Then \( x=\frac{75}{12}=6.25 \)

If two triangles \( \triangle MNP\cong\triangle VWX \), then their corresponding sides are equal. The shortest side of \( \triangle MNP \) is \( \overline{PM} \). In congruent triangles, corresponding sides are equal. The side corresponding to \( \overline{PM} \) in \( \triangle VWX \) is \( \overline{XV} \) (because the order of the letters in the congruence statement \( \triangle MNP\cong\triangle VWX \) gives the correspondence \( M
ightarrow V \), \( N
ightarrow W \), \( P
ightarrow X \), so \( PM \) corresponds to \( XV \)).

Step1: Find the similarity ratio

The original triangle has sides 8, 15, 17. The longest side is 17. The similar triangle has the longest side 51. The similarity ratio \( r=\frac{51}{17} = 3 \)

Step2: Find the shortest side of the similar triangle

The shortest side of the original triangle is 8. For similar triangles, the sides are in proportion. So the shortest side of the similar triangle is \( 8\times r=8\times3 = 24 \)

Answer:

6.25

Question 2