Question Video: Finding the Unknown Side Lengths in a Triangle given the Side Lengths of a Similar One Mathematics

Video Transcript

Given that 𝐴𝐵 equals 24 centimeters, 𝐴𝐷 equals 36 centimeters, 𝐴𝐶 equals 18 centimeters, and 𝐸𝑌 equals 15 centimeters, find the length of line segment 𝐴𝐸 and line segment 𝐷𝑋.

Starting by filling in the lengths on the diagram, we can also see that we have three parallel lines marked on this diagram. And we also have two transversals. A transversal is a line which passes through two lines in the same plane at two distinct points. We can use the fact that if three or more parallel lines are cut by two transversals, then they divide the transversals proportionally.

As we’re given the lengths on the line 𝐷𝐵, we can use this to work out the proportional relationship between the segments. We can write that the segment 𝐷𝐴 over 𝐴𝐵 is equal to the segment 𝐴𝐸 over 𝐴𝐶. And we can then substitute the values that we’re given for each segment. On the left-hand side, we have the values of 36 over 24, which is equal to 𝐴𝐸 over 18.

To find the missing value for 𝐴𝐸, we will take the cross product. Noticing that 12 is a factor of both 36 and 24 means that we can simplify the fraction on the left-hand side as three over two. And that’s equal to 𝐴𝐸 over 18. Taking the cross product then, we have three times 18 is equal to two times 𝐴𝐸. So 54 equals two times 𝐴𝐸. Dividing both sides by two will give us that 27 equals 𝐴𝐸. And therefore, 𝐴𝐸 is equal to 27 centimeters. And we found our first missing length.

To find the next missing length of 𝐷𝑋, we can notice that, on the other transversal, the length 𝐸𝑌 will be the corresponding length. Defining the length 𝐷𝑋 as 𝑎, we can then write that 𝑎 over 24 is equal to 15 over 18. We can simplify the fraction 15 over 18 to give us 𝑎 over 24 equals five-sixths. We can then take the cross product. So 𝑎 times six or six 𝑎 is equal to 24 times five. And simplifying, we have six 𝑎 equals 120. So 𝑎 equals 20. And given that we define the length 𝐷𝑋 as 𝑎, then we have 𝐷𝑋 equals 20 centimeters.

So our final answer is 𝐴𝐸 equals 27 centimeters. 𝐷𝑋 equals 20 centimeters.