Using the Converse of the Pythagorean Theorem, explain whether a triangle with sides 15, 10, and 12 is a right triangle.(1 point)
The converse of the Pythagorean theorem states that if in a triangle, the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
In this case, we have a triangle with sides 15, 10, and 12.
Using the converse of the Pythagorean theorem, we need to check if the square of the length of one side is equal to the sum of the squares of the lengths of the other two sides.
Let's check for all three sides:
- For side length 15, the sum of the squares of the other two sides is 10^2 + 12^2 = 100 + 144 = 244. But 15^2 = 225. Since 244 is not equal to 225, the triangle is not a right triangle.
- For side length 10, the sum of the squares of the other two sides is 12^2 + 15^2 = 144 + 225 = 369. But 10^2 = 100. Since 369 is not equal to 100, the triangle is not a right triangle.
- For side length 12, the sum of the squares of the other two sides is 10^2 + 15^2 = 100 + 225 = 325. But 12^2 = 144. Since 325 is not equal to 144, the triangle is not a right triangle.
Therefore, based on the converse of the Pythagorean theorem, the triangle with sides 15, 10, and 12 is not a right triangle.
To determine whether a triangle with sides 15, 10, and 12 is a right triangle, we can utilize the converse of the Pythagorean theorem. According to the converse, if the square of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
In this case, let's find the longest side of the triangle. We can observe that side 15 is the longest side.
Using the theorem's converse, we need to check if 15^2 is equal to 10^2 + 12^2.
Calculating this, we have:
225 = 100 + 144.
Since 225 is equal to 244, the converse of the Pythagorean theorem is not satisfied. Therefore, we can conclude that the triangle with sides 15, 10, and 12 is not a right triangle.
To determine whether a triangle is a right triangle, we need to check if the square of the length of the longest side is equal to the sum of the squares of the other two sides. This statement is known as the Converse of the Pythagorean Theorem.
For the given triangle with sides 15, 10, and 12, we need to find the longest side first. In this case, the longest side is 15.
Now, let's square the lengths of the other two sides and add them together:
10² + 12² = 100 + 144 = 244.
Next, we need to square the length of the longest side:
15² = 225.
Since the sum of the squares of the other two sides (244) is not equal to the square of the longest side (225), we can conclude that the triangle with sides 15, 10, and 12 is not a right triangle.
In summary, to determine whether a triangle is a right triangle using the Converse of the Pythagorean Theorem, we calculate the squares of the lengths of the sides and check if the sum of the squares of the two shorter sides is equal to the square of the longest side. If they are equal, the triangle is a right triangle; otherwise, it is not.