In triangle 𝐴𝐵𝐶, 𝐴𝐷 is perpendicular to 𝐵𝐶, 𝐷 lies between 𝐵 and 𝐶, 𝐵𝐷 equals eight, 𝐶𝐷 equals two, and 𝐴𝐷 equals four. Is 𝐴𝐵𝐶 a right-angled triangle?
In order for triangle 𝐴𝐵𝐶 to be a right triangle, it needs to have a right angle at 𝐶𝐴𝐵. Triangle 𝐴𝐶𝐷, the smaller triangle, is inscribed inside the larger one. Something that we know about inscribed triangles is that if an altitude is drawn from the right angle of any right triangle, in our case that’s the big triangle 𝐴𝐵𝐶, then the two triangles formed are similar to the original triangle. And all three triangles are similar to each other. We have our small triangle, triangle 𝐴𝐷𝐶, and our large triangle 𝐴𝐵𝐶. If we can prove that these two triangles are similar to each other, then we can prove that triangle 𝐴𝐵𝐶 is a right triangle.
Let’s start with what we know. 𝐴𝐷 is perpendicular to 𝐵𝐶. That’s here, the right angle. Point 𝐷 lies between 𝐵 and 𝐶. 𝐵𝐷 equals eight. 𝐶𝐷 equals two. And 𝐴𝐷 equals four. In order to prove similarity, we need to solve for a missing side, side 𝐴𝐶.
Side 𝐴𝐶 is the hypotenuse of the smaller triangle. That means we can use 𝑎 squared plus 𝑏 squared equals 𝑐 squared to find the length of 𝑥. Two squared plus four squared equals 𝑐 squared, well, in our case 𝑥 squared. Four plus 16 equals 𝑥 squared. 20 equals 𝑥 squared. And 𝑥 equals the square root of 20. We could simplify the square root of 20. But at this stage, we don’t need to. So we’ll just leave it as the square root of 20.
Now that we have the side lengths, we can set up some proportions. We need the proportion of two sides in the large triangle, triangle 𝐴𝐵𝐶, to be equal to the small triangle, triangle 𝐴𝐷𝐶. The proportion we wanna set up is the side length of the small side over the hypotenuse in the large triangle must be equal to the ratio of the smallest side over the hypotenuse in the small triangle.
The shortest side in our large triangle is equal to the square root of 20. And the hypotenuse of our larger triangle is two plus eight. It’s the distance from 𝐵 to 𝐶. The ratio of side lengths in the larger triangle is the square root of 20 over 10.
Now, for the smaller triangle, triangle 𝐴𝐷𝐶, its shortest side length is side length 𝐶𝐷, which measures two. And its hypotenuse, its value across from its right angle, is the square root of 20. The ratio of side lengths for the smaller triangle is equal to two over the square root of 20.
But how do we know if the square root of 20 over 10 is equal to two over the square root of 20? Well, we can cross multiply. Is the square root of 20 times the square root of 20 equal to two times 10? The square root of 20 times itself equals 20. And two times 10 equals 20. 20 does equal 20. This proves that triangle 𝐴𝐵𝐶 is similar to triangle 𝐴𝐷𝐶. Therefore, triangle 𝐴𝐵𝐶 must be a right triangle.
The answer to the question “is 𝐴𝐵𝐶 a right-angled triangle” is yes.