### Video Transcript

Is triangle πΈπ΄π· a right-angled triangle at π΄?

Letβs begin by thinking about what we know about right-angled triangles. And as we can see that in the diagram, weβve been given the lengths of some of the sides. Letβs think of what we know about the relationship between the sides in a right-angled triangle.

The Pythagorean theorem tells us that in a right-angled triangle the sum of the squares of the two shorter sides is equal to the square of the hypotenuse, which is the longest side of a right-angled triangle. The converse of this is also true, which means that if the sum of the squares of the two shorter sides of a triangle is equal to the square of its longest side, then that triangle is right angled.

Now, the question asks us whether triangle πΈπ΄π· is right angled at π΄, which means if there is a right angle in this triangle, itβs going to be at π΄. And therefore, this makes the side opposite the right angle, so thatβs πΈπ·, the hypotenuse. Weβre, therefore, looking to answer the question, βIs π΄πΈ squared plus π΄π· squared, thatβs the sum of the squares of the two shorter sides, equal to πΈπ· squared?β

Now, given the lengths of both π΄πΈ and πΈπ·, theyβre 15.6 and 19.1 centimeters, respectively. But we donβt know the length of π΄π·. So in order to answer this question, we need to calculate it. To do so, we can look at the other triangle in the diagram, triangle π΄π΅πΆ.

Now, we do know that this triangle is right angled. And weβve been given the length of its two shorter sides, which means we can apply the Pythagorean theorem to find the length of its hypotenuse, π΄πΆ. By the Pythagorean theorem then, we have that π΄πΆ squared is equal to 13.3 squared plus 15.6 squared.

Notice also these lines on the diagram, which tell us that the point π· divides the hypotenuse π΄πΆ exactly in half as the line segments π΄π· and π·πΆ are equal in length. This means that if we can calculate the length of π΄πΆ, we can find the length of π΄π· by halving it. So now, letβs work through the calculation of π΄πΆ.

First, we evaluate 13.3 squared and 15.6 squared. Then, we add them together and it tells us the π΄πΆ squared is equal to 420.25. Now, this is π΄πΆ squared. So to find π΄πΆ, we need to square root both sides of the equation. π΄πΆ is equal to the square root of 420.25, which is 20.5.

So weβve found the length of π΄πΆ; itβs 20.5 centimeters. And therefore, to find the length of π΄π·, we can halve it. Half of 20.5 is 10.25. So we now know that the length of π΄π· is 10.25 centimeters.

Now, we can return to the question we asked ourselves earlier, which was do the sides of triangle πΈπ΄π· satisfy the Pythagorean theorem. The sum of the squares of the two shorter sides is now 15.6 squared plus 10.25 squared. And evaluating this on a calculator, we have 348.4225. The square of the hypotenuse is 19.1 squared. And again, evaluating this on a calculator, this is equal to 364.81.

We can see that these two values are not equal. And therefore, the sides of the triangle πΈπ΄π· do not satisfy the Pythagorean theorem, which means triangle πΈπ΄π· is not a right-angled triangle.

So we can answer βnoβ to the question. And our reasoning for this is that π΄πΈ squared plus π΄π· squared is not equal to πΈπ· squared. The three sides do not satisfy the Pythagorean theorem.