Question Video: Finding Unknown Lengths in a Triangle Using the Similarity between Triangles Mathematics

Video Transcript

Find the value of 𝑥.

Here, we have a large triangle 𝐴𝐵𝐶, and inside that triangle are two smaller triangles. We’re trying to find the value of 𝑥. And the only side length that has 𝑥 in it is side 𝐷𝐶, so we need to decide what to set that equal to. The other two side lengths they give us is 22 centimetres for side 𝐴𝐷 and 22 centimetres for side 𝐷𝐵; those are congruent and that will actually be helpful. Because looking at triangle 𝐴𝐷𝐵, now we can see that this triangle is isosceles. And in an isosceles triangle, the base angles are congruent.

Angle 𝐷 is going to be 90 degrees for triangle 𝐴𝐷𝐵 because the angle that is adjacent to it — right next to it — is 90 degrees. And they make a straight line, so they should add up to 180. So angle 𝐷 is 90 and this is important. In a triangle, all three angles add up to 180 degrees. So if we know angle 𝐷 for this triangle is already 90, we’ve already used 90 degrees up. So now the remainder of 90 degrees needs to be split evenly between angle 𝐴 and angle 𝐵, forcing each of them to be 45 degrees.

The reason this is helpful is looking at angle 𝐵. Angle 𝐵 we can see that some of it is 45 degrees and the other part of it we don’t know. However, we know the entire angle 𝐵 for the large triangle is 90 degrees. So if the portion that goes with the blue triangle, triangle 𝐴𝐷𝐵, is 45 degrees, this portion must also be 45 degrees because 45 plus 45 is 90.

So now focusing on triangle 𝐶𝐷𝐵, we have a 90-degree angle, a 45-degree angle, and inside, a triangle all angles add to 180 degrees. This forces angle 𝐶 to be 45 degrees as well. Because 45 degrees plus 45 degrees plus 90 degrees is equal to 180 degrees. So the two 45 degrees are equal. This makes it an isosceles triangle. These are the base angles and the sides across from the base angles are the legs, which are congruent.

So if side 𝐷𝐵 is 22 centimetres, then side 𝐷𝐶 is 22 centimetres. Therefore, we can set four 𝑥 plus three equal to 22. The first step to solve for 𝑥 would be to subtract three from both sides of the equation. On the left, the threes cancel and on the right 22 minus three is 19. Now I need to divide both sides by four. On the left, the fours cancel and on the right, we get 19 fourths. This means the value of 𝑥 is equal to 19 fourths.