Question Video: Finding the Side Length of a Triangle Using the Similarity of Triangles Mathematics

If 𝐸𝐢 = 19 cm, determine the length of 𝐸𝐷 rounded to the nearest hundredth, if necessary.


Video Transcript

If 𝐸𝐢 equals 19 centimeters, determine the length of 𝐸𝐷 rounded to the nearest hundredth, if necessary.

We can fill in the information on the diagram that 𝐸𝐢 equals 19 centimeters. We can also establish that the length 𝐸𝐷 that we wish to find out is on the top part of this diagram. It might not look particularly obvious how we would find this length 𝐸𝐷, but there is one way we could check. If our smaller triangle 𝐸𝐢𝐴 is similar to the larger triangle of 𝐸𝐷𝐡, then that would give us a way to find the missing length of 𝐸𝐷.

Let’s have a look at both of these triangles and jot down anything that we know about the angles. From the markings on the diagram, we can see that angle 𝐴𝐢𝐸 in triangle 𝐸𝐢𝐴 is equal to angle 𝐡𝐷𝐸 in triangle 𝐸𝐷𝐡. We’re not given any information about the angle 𝐢𝐴𝐸. So, let’s look at the other angle, which is angle 𝐢𝐸𝐴. There is, in fact, another equal angle to this in triangle 𝐸𝐷𝐡. And it’s this one, angle 𝐷𝐸𝐡. We know this because these two angles are vertically opposite.

What we’ve found now then is that there are two pairs of corresponding angles congruent in our two triangles. We can then write that triangle 𝐸𝐢𝐴 and triangle 𝐸𝐷𝐡 are similar using the 𝐴𝐴 similarity rule, where the 𝐴𝐴 stands for two pairs of corresponding angles congruent.

So, how does this help us find our missing length of 𝐸𝐷? Well, in similar triangles, the corresponding sides are in proportion. So, we need to find a pair of corresponding sides and work out the proportion. We’re given the length of a pair of corresponding sides, the length 𝐷𝐡 and the length 𝐴𝐢. So, if we were going from the smaller triangle 𝐸𝐢𝐴 to the larger triangle 𝐸𝐷𝐡, we could find the proportion of sides or the scale factor by working out the new length divided by the original length.

So, our scale factor would be our new length, in this case, the length of 𝐡𝐷, over the length of 𝐴𝐢, which is nine. We could simplify this fraction to give us a scale factor of two, which means that every length on the smaller triangle is multiplied by two to give the corresponding length on the larger triangle. We now need to work out which length corresponds to the length of 𝐸𝐷 that we wish to find out. And it’s this length, 𝐢𝐸.

To find the length of 𝐸𝐷 then, we take the corresponding length on triangle 𝐸𝐢𝐴, which is 𝐢𝐴. And it’s 19 centimeters. And we multiply it by the scale factor of two. And 19 multiplied by two will give us 38 because 10 times two is 20 and nine times two is 18. Adding those would give us 38. The length units here will still be in centimeters. So, we found the length of 𝐸𝐷 as 38 centimeters. And we did this firstly by proving that the two triangles were similar.

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