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Bot GPT 4-o mini To determine if Malik can prove that the two right triangles are congruent, we first need to examine the given information:
1. Both triangles have a hypotenuse of 13 units.
2. The first triangle has one leg measuring 12 units.
3. The second triangle has one leg measuring 5 units.
To explore the congruence, we can use the Pythagorean Theorem \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.
For the first triangle:
- One leg \(a = 12\)
- Hypotenuse \(c = 13\)
Using the Pythagorean theorem to find the other leg \(b\):
\[
12^2 + b^2 = 13^2
\]
\[
144 + b^2 = 169
\]
\[
b^2 = 169 - 144
\]
\[
b^2 = 25 \implies b = 5
\]
For the second triangle:
- One leg \(a = 5\)
- Hypotenuse \(c = 13\)
Using the Pythagorean theorem:
\[
5^2 + b^2 = 13^2
\]
\[
25 + b^2 = 169
\]
\[
b^2 = 169 - 25
\]
\[
b^2 = 144 \implies b = 12
\]
Now we have both triangles:
- Triangle 1 has legs of \(12\) and \(5\).
- Triangle 2 has legs of \(5\) and \(12\).
Since both triangles have a hypotenuse of \(13\) and legs of \(5\) and \(12\) (in some order), we can conclude that they are congruent by the Hypotenuse-Leg theorem (HL).
Thus, the answer is: **He could apply the concept of HL and show that the two triangles are congruent.**