To determine if the two right triangles are congruent, Malik can use the Hypotenuse-Leg (HL) theorem, which states that if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
In this case, both triangles have a hypotenuse of 13 units. The first triangle has one leg of 12 units, and we can apply the Pythagorean theorem to find the other leg:
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For the first triangle (with the leg of 12 and hypotenuse of 13): \[ a^2 + b^2 = c^2 \] \[ 12^2 + b^2 = 13^2 \] \[ 144 + b^2 = 169 \] \[ b^2 = 25 \] \[ b = 5 \]
So the sides of the first triangle are 12, 5, and 13.
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The other triangle has one leg of 5 and the hypotenuse of 13. The other leg can be found similarly by applying the Pythagorean theorem: \[ a^2 + 5^2 = 13^2 \] \[ a^2 + 25 = 169 \] \[ a^2 = 144 \] \[ a = 12 \]
So the sides of the second triangle are also 5, 12, and 13.
Since both triangles have a hypotenuse of 13 and one leg of 5 (and the other leg is 12), Malik can conclude that the two right triangles are congruent by using the HL theorem.
Thus, the correct response for Malik is: He could apply the concept of HL and show that the two triangles are congruent.
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