Find the length of 𝐵𝐷.
So, we’re trying to find this length and we can do so using similar right triangles. Two right triangles are similar to each other if an acute angle in one triangle is equal to the measure of an acute angle in the other triangle. Triangle 𝐴𝐵𝐶 and triangle 𝐷𝐵𝐴 each have a right angle at angle 𝐵𝐴𝐶 and at angle 𝐵𝐷𝐴. And angle 𝐵𝐷𝐴 is equal to 90 degrees because angle 𝐴𝐷𝐶 — this one — is 90 degrees. And since they’re right next to each other and make a straight line, angle 𝐵𝐷𝐴 is 90 degrees.
So and these triangles both share a common angle measure at angle 𝐵. They share that angle. Therefore, triangle 𝐴𝐵𝐶 and triangle 𝐷𝐵𝐴 are similar and their side lengths are proportional to each other. So, we can set up a proportion: 𝐵𝐷 is to 𝐵𝐴 as 𝐴𝐵 is to 𝐶𝐵.
And we can solve proportions by cross multiplying and solving. So, we have 𝐵𝐷 times 𝐶𝐵 is equal to 𝐵𝐴 times 𝐴𝐵. But those are the exact same side. So, we can write it as 𝐴𝐵 squared.
So, let’s begin plugging in some side lengths. We know that 𝐴𝐵 is equal to 15 centimeters. We don’t know 𝐶𝐵. And 𝐵𝐷 is actually what we’re trying to solve for. So, is there a way that we can find the length of 𝐶𝐵? We can, by using the Pythagorean theorem because triangle 𝐴𝐵𝐶 is a right triangle.
The Pythagorean theorem states the square of the longest side is equal to the sum of the squares of the shorter sides. So, out of triangle 𝐴𝐵𝐶, the longest side is across from the 90-degree angle, which would be 𝐶𝐵, making the two shorter sides 15 and 20. 15 squared is 225 and 20 squared is 400. Adding them together, we get 625.
And to solve for 𝐶𝐵, we need to square root both sides. Therefore, 𝐶𝐵 is equal to 25. And now we can plug this in. So, let’s go ahead and square 15. And we get 225.
Now to solve for 𝐵𝐷, we need to divide both sides of the equation by 25. The 25s cancel on the left. And we find that 𝐵𝐷 is equal to nine. Therefore, the length of 𝐵𝐷 is equal to nine centimeters.