Using the converse of the Pythagorean theorem, explain whether a triangle with sides 15, 10, and 12 is a right triangle.
A. No, because 244 is not equal to 225
B. No, because 369 is not equal to 100
C. No, because 325 is not equal to 144
D. No, because 22 is not equal to 15
The correct answer is D. No, because 22 is not equal to 15.
According to the converse of the Pythagorean theorem, if the square of the length of the longest side of a triangle is not equal to the sum of the squares of the other two sides, then the triangle is not a right triangle.
In this case, using the converse of the Pythagorean theorem, we calculate that 15^2 + 10^2 = 225 + 100 = 325, which is not equal to the square of the length of the longest side, 12^2 = 144. Therefore, the triangle with sides 15, 10, and 12 is not a right triangle.
Elaborate?
Sure! Let's elaborate further.
The Pythagorean theorem states that in a right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side (hypotenuse). Mathematically, it can be written as:
a^2 + b^2 = c^2
where a and b are the lengths of the legs, and c is the length of the hypotenuse.
Now, the converse of the Pythagorean theorem states that if in a triangle, the sum of the squares of the lengths of two sides is equal to the square of the length of the third side, then the triangle is a right triangle.
In this case, we have a triangle with sides 15, 10, and 12. To determine if it is a right triangle, we need to check if the sum of the squares of the two shorter sides (15 and 10) is equal to the square of the longest side (12).
Using the Pythagorean theorem, we find that 15^2 + 10^2 = 225 + 100 = 325. However, this is not equal to the square of the longest side (12^2 = 144).
Since the sum of the squares of the two shorter sides is not equal to the square of the longest side, we can conclude that the triangle with sides 15, 10, and 12 is not a right triangle, according to the converse of the Pythagorean theorem.
Therefore, the correct answer is D. No, because 22 is not equal to 15.
To determine if a triangle is a right triangle using the converse of the Pythagorean theorem, we need to check if the square of the longest side is equal to the sum of the squares of the other two sides.
In this case, the longest side is 15 and the other two sides are 10 and 12. To find out if it's a right triangle, we need to calculate the squares of each side and see if they satisfy the equation.
The square of 15 is 225 (15^2). Now we need to find the sum of the squares of the other two sides.
The square of 10 is 100 (10^2).
The square of 12 is 144 (12^2).
If the converse of the Pythagorean theorem holds true, the sum of the squares of the other two sides should be equal to the square of the longest side.
Let's check:
100 + 144 = 244
Since 244 is not equal to 225, we can conclude that the triangle with sides 15, 10, and 12 is not a right triangle.
Therefore, the correct answer is A. No, because 244 is not equal to 225.