### Video Transcript

Is triangle π΄πΆπ· a right triangle at πΆ?

Essentially, we want to know if angle πΆ inside of triangle π΄πΆπ· 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 letβs go ahead and try to replace pieces of the converse of the Pythagorean theorem with information from triangle π΄πΆπ·. So what are the two shorter sides and what is the longer side? Well, angle πΆ is supposed to be the ninety degree angle. So DA would be the longer side which would make π·πΆ and π΄πΆ the shorter sides. So we can say if π·πΆ squared plus π΄πΆ squared equals DA squared, then triangle π΄πΆπ· is a right triangle at πΆ. So letβs plug in the information that we have.

We know that π·πΆ is equal to thirty-four. We donβt know the length of π΄πΆ. And DA can be replaced with forty-four. So what we need to do is find what the value of π΄πΆ squared would be. Once we find where π΄πΆ squared is equal to, we can plug it in. And if our question is true, then we can state that triangle π΄πΆπ· is a right triangle at πΆ.

We can find the value of π΄πΆ 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 π΄π΅πΆ, π΄π΅ and πΆπ΅ are the shorter sides and π΄πΆ is the longer side because itβs across from the ninety degree angle. So letβs 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 πΆπ΅ squared plus π΄π΅ squared is equal to π΄πΆ squared. And keep in mind, π΄πΆ squared is what we need in order to solve this problem. We can replace πΆπ΅ with twenty-one and π΄π΅ with seventeen point six.

So to solve for π΄πΆ 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 π΄πΆ squared is equal to seven hundred and fifty point seven six. And thatβs exactly what we need.

That way we can decide if triangle π΄πΆπ· is a right triangle at πΆ. So letβs replace π΄πΆ 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 letβs 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 πΆ.

So our answer is: No, triangle π΄πΆπ· is not a right triangle at πΆ.