To determine whether the two right triangles are congruent, Malik would most effectively use the concept of HL (Hypotenuse-Leg) congruence for right triangles. Given that both triangles have a hypotenuse of 13 units, and one leg of the first triangle is 12 units while the second triangle has a leg of 5 units, Malik can find the missing leg of both triangles using the Pythagorean Theorem:
For the first triangle:
- Hypotenuse = 13
- One leg = 12
- Let the other leg be \( b \).
Using the Pythagorean theorem: \[ a^2 + b^2 = c^2 \] \[ 12^2 + b^2 = 13^2 \] \[ 144 + b^2 = 169 \] \[ b^2 = 169 - 144 \] \[ b^2 = 25 \] \[ b = 5 \]
Now we have the sides of the first triangle as 5 and 12 units.
For the second triangle:
- Hypotenuse = 13
- One leg = 5
- Let the other leg be \( a \).
Using the Pythagorean theorem: \[ a^2 + 5^2 = 13^2 \] \[ a^2 + 25 = 169 \] \[ a^2 = 169 - 25 \] \[ a^2 = 144 \] \[ a = 12 \]
Now we have the sides of the second triangle as 5 and 12 units.
Since both triangles have a hypotenuse of 13 units and one leg measuring 12 units while the other leg measures 5 units, Malik can conclude that the two triangles are congruent by the HL theorem.
Thus, the most appropriate response is: He could apply the concept of HL and show that the two triangles are congruent.
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