Question

which statement about the two triangles shown below is true?

triangle jkl with sides: jk = 6, kl = 12, jl = 15. triangle mnp with sides: mn = 8, np = 4, mp = 10.

choose the correct answer.

a) the triangles are not similar.

b) the triangles are similar by aa similarity.

c) the triangles are similar by sas similarity.

d) the triangles are similar by sss similarity.

Explanation:

Step1: Identify side lengths

Triangle \( JKL \): \( JK = 6 \), \( KL = 12 \), \( JL = 15 \)
Triangle \( MNP \): \( MN = 8 \), \( NP = 4 \), \( MP = 10 \) (Wait, correct labels: \( MN = 8 \), \( NP = 4 \), \( MP = 10 \)? Wait, original: \( M \) to \( P \) is 8? No, original: \( M \) to \( P \) is 8? Wait, the triangles: \( \triangle JKL \) has sides 6, 12, 15; \( \triangle MNP \) (wait, labels: \( M \), \( P \), \( N \)): \( MP = 8 \), \( PN = 4 \), \( MN = 10 \)? Wait, no, the second triangle: \( M \) to \( P \) is 8, \( P \) to \( N \) is 4, \( M \) to \( N \) is 10. So sides: 8, 4, 10.

Step2: Check ratios of corresponding sides

Arrange sides in order (smallest to largest):
\( \triangle JKL \): 6, 12, 15 (sorted: 6, 12, 15)
\( \triangle MNP \): 4, 8, 10 (sorted: 4, 8, 10)

Now check ratios:
\( \frac{4}{6} = \frac{2}{3} \), \( \frac{8}{12} = \frac{2}{3} \), \( \frac{10}{15} = \frac{2}{3} \).

All three ratios are equal (\( \frac{2}{3} \)), so by SSS (Side - Side - Side) Similarity Criterion, the triangles are similar. Wait, but let's confirm the correspondence. Wait, 4 corresponds to 6, 8 to 12, 10 to 15. All ratios \( \frac{4}{6}=\frac{8}{12}=\frac{10}{15}=\frac{2}{3} \). So SSS similarity. Wait, but let's check the options. Option D: "The triangles are similar by SSS Similarity."

Wait, but wait: maybe I mixed up the labels. Let's re - check:
\( \triangle JKL \): \( JK = 6 \), \( KL = 12 \), \( JL = 15 \)
\( \triangle MPN \): \( MP = 8 \), \( PN = 4 \), \( MN = 10 \). Wait, no, the second triangle: \( M \) to \( P \) is 8, \( P \) to \( N \) is 4, \( M \) to \( N \) is 10. So sides: 8, 4, 10. So sorted: 4, 8, 10. \( \triangle JKL \) sorted: 6, 12, 15. Ratios: 4/6 = 2/3, 8/12 = 2/3, 10/15 = 2/3. So all three sides are proportional. Thus, SSS similarity. So the correct answer is D.

Answer:

D. The triangles are similar by SSS Similarity.