Video Transcript
If 𝐻𝐷 equals 17.5 centimeters, 𝐷𝐴 equals 11.3 centimeters, and 𝐶𝐵 equals 70 centimeters, find the length of 𝐴𝐶.
Let’s begin by filling in the given measurements onto the diagram. So we have 𝐻𝐷 is 17.5 centimeters, 𝐷𝐴 is 11.3 centimeters, and 𝐶𝐵 equals 70 centimeters. The length that we want to calculate is 𝐴𝐶. It might be useful to see if we can establish if the triangle 𝐴𝐷𝐻 and 𝐴𝐶𝐵 are similar. Two of the ways that we can show similarity are by using the AA rule or the SSS rule. In the AA rule, we show that there are two pairs of corresponding angles congruent. In the SSS rule, we show that there are three pairs of corresponding sides in proportion.
In this question, we can see that we’re not given enough information about the sides. So let’s see if we can use the AA rule. We’re not given any angle measurements. But if we look at this angle, 𝐻𝐴𝐷, there’s a congruent angle to it. And that’s at the angle 𝐶𝐴𝐵. This is because we have a pair of vertically opposite angles. Looking at the angle 𝐴𝐻𝐷 and using the fact that we have a pair of parallel lines and a transversal 𝐻𝐵, then the angle 𝐴𝐵𝐶 would be congruent to this one. This means that we’ve found two pairs of corresponding angles congruent. And it’s sufficient to say that our triangles 𝐴𝐻𝐷 and 𝐴𝐵𝐶 are similar.
Notice that we could’ve also used the angles 𝐻𝐷𝐴 and 𝐴𝐶𝐵 to show another pair of corresponding congruent angles. In a triangle, knowing that two pairs of corresponding angles are congruent automatically means that the final pair of corresponding angles are also congruent. So now that we’ve shown that we have similar triangles, let’s see if we can work out the length of 𝐴𝐶.
In similar triangles, the sides are in proportion, so let’s see if we can work out this proportion. We’re given the lengths of 𝐶𝐵 and 𝐻𝐷, and these two sides are corresponding. We want to work out the length 𝐴𝐶, so we’ll need to establish which side is corresponding to this one. Well, it’s the length 𝐴𝐷. When we’re writing our proportion relationship, we want to make sure that we get our lengths 𝐴𝐶 and 𝐴𝐷 in the correct place. 𝐴𝐶 is part of the triangle that also includes 𝐶𝐵. And 𝐴𝐷 is part of the triangle that includes the length 𝐻𝐷.
We can now fill in the lengths that we know. 𝐶𝐵 is 70 centimeters, 𝐻𝐷 is 17.5 centimeters, and 𝐴𝐷 is 11.3 centimeters. We can take the cross-product to find our missing length for 𝐴𝐶. This gives us 𝐴𝐶 times 17.5 equals 70 times 11.3. Evaluating the right-hand side gives us 17.5 times 𝐴𝐶 equals 791. Dividing both sides by 17.5 gives us that 𝐴𝐶 equals 45.2. And the units here will be centimeters.
An alternative method of working out could have included that the scale factor from the triangle 𝐴𝐷𝐻 to triangle 𝐴𝐵𝐶 is four. So we would multiply the lengths by four. Multiplying 𝐴𝐷, which is 11.3 centimeters, by four would’ve given us that 𝐴𝐶 is equal to 45.2 centimeters. Either method would lead us to the answer that 𝐴𝐶 is 45.2 centimeters.
