Video: Checking Whether the Given Triangle Is Right-Angled or Not given Its Side Lengths

Is β³ πΈπ΄π· a right-angled triangle at π΄?

04:02

Video Transcript

Is triangle πΈπ΄π· a right-angled triangle at π΄?

Letβs begin by thinking about what we know about right-angled triangles. And as we can see that in the diagram, weβve been given the lengths of some of the sides. Letβs 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 πΈπ΄π· is right angled at π΄, which means if there is a right angle in this triangle, itβs going to be at π΄. And therefore, this makes the side opposite the right angle, so thatβs πΈπ·, the hypotenuse. Weβre, therefore, looking to answer the question, βIs π΄πΈ squared plus π΄π· squared, thatβs the sum of the squares of the two shorter sides, equal to πΈπ· squared?β

Now, given the lengths of both π΄πΈ and πΈπ·, theyβre 15.6 and 19.1 centimeters, respectively. But we donβt 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΅πΆ.

Now, we do know that this triangle is right angled. And weβve 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΄πΆ. By the Pythagorean theorem then, we have that π΄πΆ 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΄πΆ exactly in half as the line segments π΄π· and π·πΆ are equal in length. This means that if we can calculate the length of π΄πΆ, we can find the length of π΄π· by halving it. So now, letβs work through the calculation of π΄πΆ.

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

So weβve found the length of π΄πΆ; itβs 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 πΈπ΄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 πΈπ΄π· do not satisfy the Pythagorean theorem, which means triangle πΈπ΄π· is not a right-angled triangle.

So we can answer βnoβ to the question. And our reasoning for this is that π΄πΈ squared plus π΄π· squared is not equal to πΈπ· squared. The three sides do not satisfy the Pythagorean theorem.