What we see: a Two-Part Analysis geometry question. Triangle JKL is a right triangle, KL = 60, and JK is longer than JL. We pick one length for JK and one for JL from 24, 30, 36, 48, 60, 90.
Step 1. Decide which side can be the long side (the hypotenuse).
What we see: we are not told where the right angle is.
What it means: the hypotenuse is KL, JK or JL. It cannot be JL, because JK is longer than JL and the hypotenuse is always the longest side. So the hypotenuse is KL or JK.
What we do and why: we test both cases with the Pythagorean rule, \(\text{leg}^2 + \text{leg}^2 = \text{hypotenuse}^2\).
Step 2. Case 1: KL = 60 is the hypotenuse.
What we see: \(JK^2 + JL^2 = 60^2 = 3{,}600\), with both sides shorter than 60.
What it means: 60 is \(12 \times 5\), so a 3-4-5 triangle scaled by 12 fits: \(36, 48, 60\).
What we do and why: check \(48^2 + 36^2 = 2{,}304 + 1{,}296 = 3{,}600\). It works, and JK = 48 is longer than JL = 36, as required.
Step 3. Case 2: JK is the hypotenuse.
What we see: then \(JK^2 = JL^2 + 3{,}600\), so JK must be longer than 60. The only choice above 60 is 90.
What it means: \(JL^2 = 8{,}100 - 3{,}600 = 4{,}500\), and 4,500 is not a perfect square, so no choice works.
What we do and why: this case gives nothing, so Case 1 is the only answer.
The trap: JK = 60 or JK = 90. People see 60 and assume it must be a leg and JK the hypotenuse, then hunt for a pair that cannot exist in the list.
Faster way: when a side is a multiple of 5, try the 3-4-5 triangle first. 60 = 5 × 12 gives 36 and 48 at once, and both are in the list.
Takeaway: when the right angle is not named, test each side that could be the hypotenuse; "longer than" clues rule some of them out fast.
Answer: JK 48, JL 36
