Question

preimage
image
what is the length of side jk in the image?
60
80
70
75
(preimage triangle: kl=60, kj=100, right angle at l; image triangle: kl=36, right angle at l, jk=?)

Explanation:

Step1: Identify similar triangles

The preimage and image are right triangles, so they are similar (by AA similarity, as both have a right angle and the triangles are transformations of each other).

Step2: Find the scale factor

First, find the length of the other leg in the preimage. Using Pythagoras in preimage: \( \text{leg}_2=\sqrt{100^2 - 60^2}=\sqrt{10000 - 3600}=\sqrt{6400} = 80 \). Now, the vertical leg in preimage is 60, in image is 36. Scale factor \( k=\frac{36}{60}=\frac{3}{5} \).

Step3: Calculate JK in image

The hypotenuse in preimage is 100? Wait, no, wait. Wait, in preimage, the hypotenuse is JK? Wait, no, preimage: triangle LKJ, right - angled at L. So sides: LK = 60, LJ = 80 (from step 2), JK = 100. In image, triangle JLK, right - angled at L. The vertical side (LK in image) is 36. Since scale factor is \( \frac{3}{5} \), the hypotenuse JK in image is \( 100\times\frac{3}{5}=60 \)? Wait, no, wait. Wait, maybe I mixed up. Wait, in preimage, LK = 60 (vertical), LJ = 80 (horizontal), JK = 100 (hypotenuse). In image, the vertical side (now KL) is 36. So the scale factor is \( \frac{36}{60}=\frac{3}{5} \). The hypotenuse in image (JK) should be preimage hypotenuse (JK = 100) times scale factor? Wait, no, maybe the preimage hypotenuse is 100, and the image's hypotenuse is scaled. Wait, but let's check the other way. Wait, maybe the preimage has legs 60 and 80, hypotenuse 100. The image has leg 36, so the other leg in image is \( 80\times\frac{3}{5}=48 \)? No, the options are 60,80,70,75. Wait, maybe I made a mistake. Wait, the preimage: right triangle with legs 60 and 80 (since \( 60 - 80 - 100 \) is a Pythagorean triple). The image: right triangle with one leg 36. Let's find the scale factor. \( \frac{36}{60}=\frac{3}{5} \). Then the hypotenuse of the image should be \( 100\times\frac{3}{5}=60 \)? But wait, the options have 60 as a choice. Wait, maybe the preimage's hypotenuse is 100, and the image's hypotenuse is 60. Alternatively, maybe the preimage's leg is 60, image's leg is 36, scale factor \( 36/60 = 3/5 \). The other leg in preimage is 80, so in image it's \( 80\times3/5 = 48 \), but we need JK. Wait, maybe the preimage's JK is 100, and image's JK is \( 100\times3/5=60 \). So that's the answer.

Answer:

60