To prove that triangles $\triangle JKL$ and $\triangle TUV$ are congruent using the Distance Formula and the Hypotenuse-Leg Theorem (HL), we need to follow these steps:

Step 1: Find the lengths of the legs and hypotenuses of both triangles.

We'll use the Distance Formula to calculate the side lengths for both triangles. The formula for distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Triangle $\triangle JKL$:

• Leg $JK$: Find the distance between points $J(x_1, y_1)$ and $K(x_2, y_2)$.
• Leg $KL$: Find the distance between points $K(x_2, y_2)$ and $L(x_3, y_3)$.
• Hypotenuse $JL$: Find the distance between points $J(x_1, y_1)$ and $L(x_3, y_3)$.

Triangle $\triangle TUV$:

• Leg $TU$: Find the distance between points $T(x_4, y_4)$ and $U(x_5, y_5)$.
• Leg $UV$: Find the distance between points $U(x_5, y_5)$ and $V(x_6, y_6)$.
• Hypotenuse $TV$: Find the distance between points $T(x_4, y_4)$ and $V(x_6, y_6)$.

Step 2: Verify if the legs and hypotenuses are equal.

We are interested in verifying the following:

• Legs are congruent: $JK = TU$ and $KL = UV$
• Hypotenuses are congruent: $JL = TV$

Step 3: Apply the Hypotenuse-Leg Theorem (HL).

The Hypotenuse-Leg Theorem states that if a right triangle has a congruent hypotenuse and one congruent leg, the triangles are congruent. So, we need to verify:

• Both triangles are right triangles (which they appear to be based on the grid).
• One leg and the hypotenuse of each triangle are congruent.

If these conditions hold, then by the HL Theorem, the two triangles are congruent.

I will now calculate the distances. Let's proceed with the calculations for the distance formula using the coordinates provided in the image. I’ll need the specific coordinates of the points $J, K, L, T, U, V$ from the grid.

Could you provide the coordinates or confirm the locations of these points on the grid?