Which lengths represent the side lengths of a right triangle? Use pencil and graph paper. Draw each triangle to verify your answer. Triangle 1: 12, 9, 21 Triangle 2: 9, 12, 15 Triangle 3: 15, 36 , 39.
Answer Choices:
A. Triangle 2 and Triangle 3 are right triangles.
B. Triangle 1 and Triangle 3 are right triangles.
C. All of the triangles are right triangles.
D. None of the triangles are right triangles.
use a^2 + b^2 = c^2 to check
You can save yourself some calculation by learning a few of the basic Pythagorean triples, such as
3-4-5, 5-12-13, 8-15-17, 7-24-25
and their multiples.
To determine if a triangle is a right triangle, we need to check if it satisfies the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
Let's calculate using pencil and graph paper:
Triangle 1: Side lengths: 12, 9, 21
Let's check using the Pythagorean theorem:
12^2 + 9^2 = 144 + 81 = 225
21^2 = 441
The sum of the squares of the two shorter sides is not equal to the square of the longest side. So, Triangle 1 is not a right triangle.
Triangle 2: Side lengths: 9, 12, 15
Using the Pythagorean theorem:
9^2 + 12^2 = 81 + 144 = 225
15^2 = 225
The sum of the squares of the two shorter sides is equal to the square of the longest side. Therefore, Triangle 2 is a right triangle.
Triangle 3: Side lengths: 15, 36, 39
By applying the Pythagorean theorem:
15^2 + 36^2 = 225 + 1296 = 1521
39^2 = 1521
The sum of the squares of the two shorter sides is equal to the square of the longest side. Thus, Triangle 3 is a right triangle.
Based on our calculations:
Triangle 1 is not a right triangle.
Triangle 2 is a right triangle.
Triangle 3 is a right triangle.
Therefore, the correct answer is B. Triangle 1 and Triangle 3 are right triangles.
To determine if a triangle is a right triangle, we need to check if it satisfies the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
To find the answer, we need to calculate the squares of the side lengths and check if they satisfy the Pythagorean theorem.
Let's go through each triangle:
Triangle 1: Side lengths 12, 9, and 21
We calculate the squares of the side lengths:
12^2 = 144
9^2 = 81
21^2 = 441
According to the Pythagorean theorem:
144 + 81 = 225
225 is equal to 21^2
Since 225 is equal to the square of the hypotenuse, Triangle 1 does satisfy the Pythagorean theorem and is a right triangle.
Triangle 2: Side lengths 9, 12, and 15
We calculate the squares of the side lengths:
9^2 = 81
12^2 = 144
15^2 = 225
According to the Pythagorean theorem:
81 + 144 = 225
225 is equal to 15^2
Again, since 225 is equal to the square of the hypotenuse, Triangle 2 also satisfies the Pythagorean theorem and is a right triangle.
Triangle 3: Side lengths 15, 36, and 39
We calculate the squares of the side lengths:
15^2 = 225
36^2 = 1296
39^2 = 1521
According to the Pythagorean theorem:
225 + 1296 = 1521
1521 is equal to 39^2
Once again, since 1521 is equal to the square of the hypotenuse, Triangle 3 satisfies the Pythagorean theorem and is a right triangle.
In conclusion, both Triangle 1 and Triangle 2 satisfy the Pythagorean theorem and are right triangles. Therefore, the correct answer is:
A. Triangle 2 and Triangle 3 are right triangles.