Question

triangle jkl is similar to triangle pqr. what is the length of jl? 4 3 3.75 7.5 (with images of triangles jkl and pqr, and multiple - choice options)

Explanation:

Step1: Identify Corresponding Sides

In similar triangles, corresponding sides are proportional. Let's find the ratio of sides of triangle \( PQR \) and \( JKL \). First, identify corresponding sides. Let's assume \( KL \) corresponds to \( QR \), \( JK \) to \( PQ \), and \( JL \) to \( PR \)? Wait, no, let's check the lengths. \( KL = 2 \), \( QR = 8 \)? Wait, no, looking at the triangles: triangle \( JKL \) has side \( KL = 2 \), triangle \( PQR \) has side \( QR = 8 \)? Wait, no, the second triangle \( PQR \): \( PQ = 12 \), \( QR = 8 \), \( PR = 15 \). Triangle \( JKL \): \( KL = 2 \), let's find the ratio. Wait, maybe \( KL \) corresponds to \( QR \), so ratio of \( JKL \) to \( PQR \) is \( \frac{KL}{QR} = \frac{2}{8} = \frac{1}{4} \)? No, that doesn't match. Wait, maybe \( PQR \) is larger, so ratio of \( PQR \) to \( JKL \) is \( \frac{QR}{KL} = \frac{8}{2} = 4 \)? Then \( PR = 15 \), so \( JL \) would be \( \frac{15}{4} = 3.75 \)? No, wait, maybe I mixed up. Wait, let's list the sides:

Triangle \( JKL \): sides? Let's see, \( KL = 2 \). Triangle \( PQR \): \( PQ = 12 \), \( QR = 8 \), \( PR = 15 \).

Since they are similar, the ratio of corresponding sides should be equal. Let's find the ratio of \( PQR \) to \( JKL \). Let's take \( PQ \) and \( JK \)? Wait, no, maybe \( KL \) corresponds to \( QR \), \( JK \) corresponds to \( PQ \), \( JL \) corresponds to \( PR \).

Wait, let's check the ratio of \( QR \) (8) to \( KL \) (2): 8/2 = 4. Then \( PQ \) is 12, so \( JK \) would be 12/4 = 3? No, but we need \( JL \). \( PR \) is 15, so \( JL \) would be 15/4 = 3.75? No, that's not one of the options? Wait, no, maybe the ratio is \( JKL \) to \( PQR \) as \( 2/8 = 1/4 \), so \( JL \) (corresponding to \( PR = 15 \)) would be \( 15 \times (1/4) = 3.75 \)? But 3.75 is an option. Wait, but let's check another ratio. \( PQ = 12 \), \( QR = 8 \), \( PR = 15 \). Let's see the ratio of \( PQ \) to \( QR \) is 12/8 = 3/2. \( PR \) to \( QR \) is 15/8. Wait, maybe I got the correspondence wrong.

Wait, maybe \( KL \) corresponds to \( PQ \)? No, \( KL = 2 \), \( PQ = 12 \), ratio 2/12 = 1/6. No. Wait, maybe \( KL \) corresponds to \( PQ \)? No, that doesn't make sense. Wait, let's look at the lengths. Triangle \( JKL \) is smaller, triangle \( PQR \) is larger. Let's find the scale factor. Let's take \( QR = 8 \) and \( KL = 2 \), so scale factor from \( JKL \) to \( PQR \) is 8/2 = 4. Then \( PR = 15 \), so \( JL = 15 / 4 = 3.75 \)? But 3.75 is an option. Wait, but let's check \( PQ = 12 \), so \( JK = 12 / 4 = 3 \), which is also an option. But the question is about \( JL \). Wait, maybe the correspondence is \( KL \) (2) with \( PQ \) (12)? No, 2 and 12: ratio 1/6. Then \( PR = 15 \), so \( JL = 15 * (1/6) = 2.5 \), not an option. Wait, maybe \( KL \) corresponds to \( QR \) (8), so ratio 2/8 = 1/4. Then \( PR = 15 \), so \( JL = 15 * (1/4) = 3.75 \). Yes, 3.75 is an option. Wait, but let's check another pair. \( PQ = 12 \), so \( JK = 12 * (1/4) = 3 \), which is an option. But the question is \( JL \), so \( PR = 15 \), so \( JL = 15 * (1/4) = 3.75 \). Wait, but maybe I mixed up the correspondence. Let's try \( PQ = 12 \), \( JK = x \), \( QR = 8 \), \( KL = 2 \), \( PR = 15 \), \( JL = y \). Then \( \frac{JK}{PQ} = \frac{KL}{QR} = \frac{JL}{PR} \). So \( \frac{2}{8} = \frac{y}{15} \). Solving for \( y \): \( y = 15 * (2/8) = 15 * (1/4) = 3.75 \). Yes, that's correct. So \( JL = 3.75 \).

Step2: Verify the Ratio

Check the ratio of \( KL \) to \( QR \): \( 2/8 = 1/4 \). Then \( JL \) to \( PR \) should be \( 1/4 \), so \( JL = 15 * (1/4) = 3.75 \). That matches the calculation.

Answer:

3.75