Question
Malik is comparing 2 right triangles. Both triangles have a hypotenuse of 13 units. The first triangle has a leg of 12 units and the other triangle has a leg of 5 units. How could Malik prove that the two triangles are congruent?(1 point)
Responses
He would need additional information to prove that the two triangles are congruent.
He would need additional information to prove that the two triangles are congruent.
He can conclude that they are congruent because they are both right triangles.
He can conclude that they are congruent because they are both right triangles.
He could apply the concept of HL and show that the two triangles are congruent.
He could apply the concept of HL and show that the two triangles are congruent.
He could apply the Pythagorean Theorem to show that the two missing sides make both triangles congruent.
Responses
He would need additional information to prove that the two triangles are congruent.
He would need additional information to prove that the two triangles are congruent.
He can conclude that they are congruent because they are both right triangles.
He can conclude that they are congruent because they are both right triangles.
He could apply the concept of HL and show that the two triangles are congruent.
He could apply the concept of HL and show that the two triangles are congruent.
He could apply the Pythagorean Theorem to show that the two missing sides make both triangles congruent.
Answers
GPT-4o mini
To determine if the two right triangles are congruent, Malik can apply the concept of the Hypotenuse-Leg (HL) theorem. The HL theorem states that if two right triangles have a congruent hypotenuse and one pair of congruent legs, then the triangles are congruent.
In this case, both triangles have a hypotenuse of 13 units. For the first triangle with a leg of 12 units, we can use the Pythagorean theorem to find the missing leg:
\( a^2 + b^2 = c^2 \)
Where \( c = 13 \) and one leg (\( a \)) is 12:
\( 12^2 + b^2 = 13^2 \)
\( 144 + b^2 = 169 \)
\( b^2 = 169 - 144 \)
\( b^2 = 25 \)
\( b = 5 \)
This means the first triangle has legs of 12 units and 5 units.
For the second triangle, one leg is given as 5 units, and the hypotenuse is 13 units. The missing leg can also be calculated:
Using the same equation:
\( 5^2 + b^2 = 13^2 \)
\( 25 + b^2 = 169 \)
\( b^2 = 169 - 25 \)
\( b^2 = 144 \)
\( b = 12 \)
This triangle has the legs of 5 units and 12 units.
Since both triangles have one leg of 5 units, the other leg of 12 units, and both have a hypotenuse of 13 units, Malik can conclude that the two triangles are congruent by applying the HL theorem.
Therefore, the correct response is:
**He could apply the concept of HL and show that the two triangles are congruent.**
In this case, both triangles have a hypotenuse of 13 units. For the first triangle with a leg of 12 units, we can use the Pythagorean theorem to find the missing leg:
\( a^2 + b^2 = c^2 \)
Where \( c = 13 \) and one leg (\( a \)) is 12:
\( 12^2 + b^2 = 13^2 \)
\( 144 + b^2 = 169 \)
\( b^2 = 169 - 144 \)
\( b^2 = 25 \)
\( b = 5 \)
This means the first triangle has legs of 12 units and 5 units.
For the second triangle, one leg is given as 5 units, and the hypotenuse is 13 units. The missing leg can also be calculated:
Using the same equation:
\( 5^2 + b^2 = 13^2 \)
\( 25 + b^2 = 169 \)
\( b^2 = 169 - 25 \)
\( b^2 = 144 \)
\( b = 12 \)
This triangle has the legs of 5 units and 12 units.
Since both triangles have one leg of 5 units, the other leg of 12 units, and both have a hypotenuse of 13 units, Malik can conclude that the two triangles are congruent by applying the HL theorem.
Therefore, the correct response is:
**He could apply the concept of HL and show that the two triangles are congruent.**