### Video Transcript

Triangles π΄π΅πΆ and π΄π·πΈ are similar. Find π₯ to the nearest integer.

Here, weβre told that the larger triangle π΄π΅πΆ and the smaller triangle π΄π·πΈ 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 πΈπ· and πΆπ΅. But as we donβt know the value of π₯, we canβt use this to find a scale factor, so weβll have to use the other pair of corresponding sides, π΄πΈ and the length π΄πΆ.

To find the scale factor going from the smaller triangle π΄π·πΈ to the larger triangle π΄π΅πΆ, we can calculate the new length divided by the original length. Our scale factor will then be the length of π΄πΆ divided by π΄πΈ. The length π΄πΆ is equal to five centimeters plus two centimeters, which is seven, over the length of π΄πΈ, 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 πΈπ·, we could multiply it by seven-fifths to get the length of πΆπ΅. We could therefore write this as seven-fifths times four π₯ minus 15, which is πΈπ·, equals π₯ plus two, which is π΅πΆ. We could then solve this equation to find the value of π₯.

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 π₯ minus 15 on the left-hand side and five π₯ plus 10 on the right-hand side. We can then begin distributing our seven across the parentheses. This gives us seven times four π₯, which is 28π₯. And seven times 15 is 105, so we have 28π₯ subtract 105, which is still equal to five π₯ plus 10.

We can then subtract five π₯ from both sides, giving us 23π₯ minus 105 equals 10. We can then add 105 to both sides of the equation, giving us 23π₯ equals 115. Finally, to find the value of π₯, we can divide both sides of the equation by 23. So π₯ equals five. We were asked for π₯ to the nearest integer, but as π₯ is already an integer, then we donβt need to do any rounding. So here we have our answer. π₯ equals five.