Video Transcript
Is triangle 𝐴𝐶𝐷 a right triangle at 𝐶?
Essentially, we want to know if angle 𝐶 inside of triangle 𝐴𝐶𝐷 is a ninety degree angle. We can make this decision using the converse of the Pythagorean theorem. It states that if the sum of the squares of the shorter sides of the triangle is equal to the square of the longer side of the triangle, then the triangle is a right triangle.
So let’s go ahead and try to replace pieces of the converse of the Pythagorean theorem with information from triangle 𝐴𝐶𝐷. So what are the two shorter sides and what is the longer side? Well, angle 𝐶 is supposed to be the ninety degree angle. So DA would be the longer side which would make 𝐷𝐶 and 𝐴𝐶 the shorter sides. So we can say if 𝐷𝐶 squared plus 𝐴𝐶 squared equals DA squared, then triangle 𝐴𝐶𝐷 is a right triangle at 𝐶. So let’s plug in the information that we have.
We know that 𝐷𝐶 is equal to thirty-four. We don’t know the length of 𝐴𝐶. And DA can be replaced with forty-four. So what we need to do is find what the value of 𝐴𝐶 squared would be. Once we find where 𝐴𝐶 squared is equal to, we can plug it in. And if our question is true, then we can state that triangle 𝐴𝐶𝐷 is a right triangle at 𝐶.
We can find the value of 𝐴𝐶 squared using the other triangle in our diagram because that triangle is a right triangle, which means we can use the Pythagorean theorem. So looking at triangle 𝐴𝐵𝐶, 𝐴𝐵 and 𝐶𝐵 are the shorter sides and 𝐴𝐶 is the longer side because it’s across from the ninety degree angle. So let’s plug that into the Pythagorean theorem. The sum of the squares of the shorter sides is equal to the square of the longer side. So 𝐶𝐵 squared plus 𝐴𝐵 squared is equal to 𝐴𝐶 squared. And keep in mind, 𝐴𝐶 squared is what we need in order to solve this problem. We can replace 𝐶𝐵 with twenty-one and 𝐴𝐵 with seventeen point six.
So to solve for 𝐴𝐶 squared, we need to square twenty-one and square seventeen point six and add those together. Twenty-one squared is equal to four hundred and forty-one. And seventeen point six squared is equal to three hundred and nine point seven six. So now we need to add these numbers together, four hundred and forty-one plus three hundred and nine point seven six. So 𝐴𝐶 squared is equal to seven hundred and fifty point seven six. And that’s exactly what we need.
That way we can decide if triangle 𝐴𝐶𝐷 is a right triangle at 𝐶. So let’s replace 𝐴𝐶 squared with seven hundred and fifty point seven six. And now, thirty-four squared is equal to one thousand one hundred and fifty-six. And forty-four squared is equal to one thousand nine hundred and thirty-six. So let’s see if this equation holds true. One thousand one hundred and fifty-six plus seven hundred and fifty point seven six is equal to one thousand nine hundred and six point seven six, which is not equal to one thousand nine hundred and thirty-six. This means that since the sum of the squares of the shorter sides is not equal to the square of the longer side, then the triangle is not a right triangle at 𝐶.
So our answer is: No, triangle 𝐴𝐶𝐷 is not a right triangle at 𝐶.
