Question Video: Finding the Unknown Lengths in a Triangle given the Other Sides’ Lengths of a Similar One Mathematics • 8th Grade

Video Transcript

Find the length of line segment 𝐷𝐵 rounded to the nearest hundredth, if needed.

In this problem, we need to find one of the lengths which makes up part of the triangle. We can observe that in the figure we have two triangles. It may be useful to consider if these triangles are in fact similar triangles as this would help us establish the unknown length. We can recall that similar triangles have corresponding angles congruent and corresponding sides in proportion.

Now, sometimes, we might find it hard to visualize the two separate triangles in a figure like this. So, it can be helpful to draw another sketch of the two triangles separately. For example, here, we can see the larger triangle 𝐴𝐶𝐵 and the smaller triangle 𝐴𝐷𝐸 drawn in the same orientation so that the angles at 𝐶 and 𝐷 are in the same position.

We can write on the lengths’ information from the original diagram that 𝐴𝐷 is 10 centimeters, 𝐴𝐸 is 13 centimeters, and 𝐴𝐶 is the sum of the lengths of 13 and 12 centimeters, which is 25 centimeters. We can note from the markings on the diagram that there are two congruent angles: angle 𝐶 and angle 𝐷. And we can observe that the angle at 𝐴 is a common or shared angle. So, its measure will be equal in the two triangles.

So, let’s return to considering if the triangles are similar. Recall that one way we can prove triangles are similar is by demonstrating that all pairs of corresponding angles are congruent. In our diagram, we know that two pairs of angles are congruent. But in fact, knowing that two pairs of angles are congruent is enough to show that all three pairs of angles are congruent because we know that the angle measures in a triangle sum to 180 degrees. And since we know that the other two pairs of corresponding angles are congruent, then the remaining pair of angles in each triangle must be congruent.

So, all corresponding angles are congruent. And importantly, we can now say that these two triangles are similar. We can write this similarity relationship as triangle 𝐴𝐶𝐵 is similar to triangle 𝐴𝐷𝐸.

Now remember, we want to find the length of the line segment 𝐷𝐵. 𝐷𝐵 is part of the longer line segment 𝐴𝐵. And we know that the other part of this line segment, 𝐴𝐷, has a length of 10 centimeters. We can note that if we knew the length of the line segment 𝐴𝐵, this would help us work out the length of line segment 𝐷𝐵. Because the triangles are similar, then line segment 𝐴𝐸 in triangle 𝐴𝐷𝐸 is corresponding to line segment 𝐴𝐵. And in similar triangles, corresponding sides are in proportion. So, we need to find a pair of corresponding sides whose lengths we are given. That would be line segments 𝐴𝐶 and 𝐴𝐷. We can write that 𝐴𝐶 over 𝐴𝐷 is equal to 𝐴𝐵 over 𝐴𝐸.

We could alternatively have written these fractions or proportions with the numerators and denominators flipped. The important bit about writing the proportion is that we keep the sides on each triangle either as both on the numerators or both on the denominators. We can then fill in the lengths that we know. This gives us 25 over 10 equals 𝐴𝐵 over 13. We might simplify the left side first to five over two before multiplying both sides by 13, which gives us that 𝐴𝐵 equals 65 over two or 32.5 centimeters.

But we haven’t finished the question yet. We’ve worked out the length of the longer line segment 𝐴𝐵, but we really want to know the length of line segment 𝐷𝐵. So 𝐷𝐵 is 32.5 centimeters subtract 10 centimeters, which is 22.5 centimeters.

Therefore, by first proving that the triangles are similar, we have determined that the length of line segment 𝐷𝐵 is 22.5 centimeters. And as this value has one decimal place, we don’t need to round the answer to the nearest hundredth.