Malik could use the hypotenuse-leg (HL) theorem to prove that the two triangles are congruent. The HL theorem states that if in two right triangles, the lengths of the hypotenuses are equal and one leg is equal in length to one leg of the other triangle, then the two triangles are congruent.
In this case, both triangles have a hypotenuse of 13 units. The first triangle has a leg of 12 units, and the second triangle has a leg of 5 units. To apply the HL theorem correctly, Malik would need to determine the lengths of the missing legs of both triangles.
For the first triangle (with a leg of 12 units):
Using the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] Where \(c = 13\) and \(a = 12\): \[ 12^2 + b^2 = 13^2 \ 144 + b^2 = 169 \ b^2 = 169 - 144 \ b^2 = 25 \ b = 5 \]
For the second triangle (with a leg of 5 units), we also find the missing leg in a similar way:
If one leg is 5, let's denote the other leg as \(x\): \[ 5^2 + x^2 = 13^2 \ 25 + x^2 = 169 \ x^2 = 169 - 25 \ x^2 = 144 \ x = 12 \]
Now, we have both triangles:
- Triangle 1: legs of 12 and 5, hypotenuse of 13.
- Triangle 2: legs of 5 and 12, hypotenuse of 13.
Both triangles are congruent by the HL theorem (hypotenuse of 13 units and corresponding legs are equal). Therefore, the correct response is:
He could apply the concept of HL and show that the two triangles are congruent.
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