Is triangle 𝐸𝐴𝐷 a right-angled triangle at 𝐴?
Let’s begin by thinking about what we know about right-angled triangles. And as we can see that in the diagram, we’ve been given the lengths of some of the sides. Let’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 𝐸𝐴𝐷 is right angled at 𝐴, which means if there is a right angle in this triangle, it’s going to be at 𝐴. And therefore, this makes the side opposite the right angle, so that’s 𝐸𝐷, the hypotenuse. We’re, therefore, looking to answer the question, “Is 𝐴𝐸 squared plus 𝐴𝐷 squared, that’s the sum of the squares of the two shorter sides, equal to 𝐸𝐷 squared?”
Now, given the lengths of both 𝐴𝐸 and 𝐸𝐷, they’re 15.6 and 19.1 centimeters, respectively. But we don’t know the length of 𝐴𝐷. 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 𝐴𝐵𝐶.
Now, we do know that this triangle is right angled. And we’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, 𝐴𝐶. By the Pythagorean theorem then, we have that 𝐴𝐶 squared is equal to 13.3 squared plus 15.6 squared.
Notice also these lines on the diagram, which tell us that the point 𝐷 divides the hypotenuse 𝐴𝐶 exactly in half as the line segments 𝐴𝐷 and 𝐷𝐶 are equal in length. This means that if we can calculate the length of 𝐴𝐶, we can find the length of 𝐴𝐷 by halving it. So now, let’s work through the calculation of 𝐴𝐶.
First, we evaluate 13.3 squared and 15.6 squared. Then, we add them together and it tells us the 𝐴𝐶 squared is equal to 420.25. Now, this is 𝐴𝐶 squared. So to find 𝐴𝐶, we need to square root both sides of the equation. 𝐴𝐶 is equal to the square root of 420.25, which is 20.5.
So we’ve found the length of 𝐴𝐶; it’s 20.5 centimeters. And therefore, to find the length of 𝐴𝐷, we can halve it. Half of 20.5 is 10.25. So we now know that the length of 𝐴𝐷 is 10.25 centimeters.
Now, we can return to the question we asked ourselves earlier, which was do the sides of triangle 𝐸𝐴𝐷 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 𝐸𝐴𝐷 do not satisfy the Pythagorean theorem, which means triangle 𝐸𝐴𝐷 is not a right-angled triangle.
So we can answer “no” to the question. And our reasoning for this is that 𝐴𝐸 squared plus 𝐴𝐷 squared is not equal to 𝐸𝐷 squared. The three sides do not satisfy the Pythagorean theorem.