Video Transcript
Triangles 𝐴𝐵𝐶 and 𝐴𝐷𝐸 are similar. Find 𝑥 to the nearest integer.
Here, we’re told that the larger triangle 𝐴𝐵𝐶 and the smaller triangle 𝐴𝐷𝐸 are similar. This means we can say that the angles are congruent, and the sides are in proportion. In order to calculate what the proportion is, we can find the scale factor between the two triangles. To find the scale factor, we can use a given pair of corresponding sides. There is a pair of corresponding sides 𝐸𝐷 and 𝐶𝐵. But as we don’t know the value of 𝑥, we can’t use this to find a scale factor, so we’ll have to use the other pair of corresponding sides, 𝐴𝐸 and the length 𝐴𝐶.
To find the scale factor going from the smaller triangle 𝐴𝐷𝐸 to the larger triangle 𝐴𝐵𝐶, we can calculate the new length divided by the original length. Our scale factor will then be the length of 𝐴𝐶 divided by 𝐴𝐸. The length 𝐴𝐶 is equal to five centimeters plus two centimeters, which is seven, over the length of 𝐴𝐸, which is five centimeters. And so our scale factor is seven-fifths.
So now we know that to go from the smaller triangle to the larger triangle, we multiply by seven-fifths. So if we look at the length 𝐸𝐷, we could multiply it by seven-fifths to get the length of 𝐶𝐵. We could therefore write this as seven-fifths times four 𝑥 minus 15, which is 𝐸𝐷, equals 𝑥 plus two, which is 𝐵𝐶. We could then solve this equation to find the value of 𝑥.
We could begin by multiplying both sides of this equation by five in order to get rid of the five on the denominator, leaving us with just seven times four 𝑥 minus 15 on the left-hand side and five 𝑥 plus 10 on the right-hand side. We can then begin distributing our seven across the parentheses. This gives us seven times four 𝑥, which is 28𝑥. And seven times 15 is 105, so we have 28𝑥 subtract 105, which is still equal to five 𝑥 plus 10.
We can then subtract five 𝑥 from both sides, giving us 23𝑥 minus 105 equals 10. We can then add 105 to both sides of the equation, giving us 23𝑥 equals 115. Finally, to find the value of 𝑥, we can divide both sides of the equation by 23. So 𝑥 equals five. We were asked for 𝑥 to the nearest integer, but as 𝑥 is already an integer, then we don’t need to do any rounding. So here we have our answer. 𝑥 equals five.
