Question

Segment KJ shown below is the hypotenuse of isosceles right triangle JLK.

On a coordinate plane, triangle J L K has points (2, 4), (5, 1), (2, negative 2).

What is the length of one of the congruent legs of JLK?
3
StartRoot 18 EndRoot
6
StartRoot 72 EndRoot

Answer 1

To find the length of one of the congruent legs of the isosceles right triangle JLK, we first need to verify the positions of points J, L, and K, along with their coordinates.

Given the coordinates:
- \( J(2, 4) \)
- \( L(5, 1) \)
- \( K(2, -2) \)

We can calculate the distances between the points to identify the lengths of the sides of the triangle.

1. **Calculate the length of segment JL:**
\[
JL = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(5 - 2)^2 + (1 - 4)^2} = \sqrt{3^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18}
\]

2. **Calculate the length of segment JK:**
\[
JK = \sqrt{(2 - 2)^2 + (-2 - 4)^2} = \sqrt{0 + (-6)^2} = \sqrt{36} = 6
\]

3. **Calculate the length of segment KL:**
\[
KL = \sqrt{(5 - 2)^2 + (1 + 2)^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18}
\]

From our calculations:
- \( JL = \sqrt{18} \)
- \( JK = 6 \)
- \( KL = \sqrt{18} \)

Since JLK is an isosceles right triangle, two sides are congruent. We can see that segments JL and KL are equal (both are \(\sqrt{18}\)), forming the congruent legs of the triangle.

Thus, the length of one of the congruent legs of triangle JLK is:

\[
\sqrt{18}
\]