### Video Transcript

π΄π΅πΆ is a triangle, where π΅πΆ equals 55 centimeters, π΄πΆ minus π΄π΅ equals 13 centimeters, and the perimeter is 124 centimeters. Find the lengths of the line segments π΄πΆ and π΄π΅, giving the answers to the nearest centimeter.

Letβs begin by thinking about the notation used to describe this triangle. The triangle has been described as π΄π΅πΆ. So these are the letters corresponding to its three vertices. Weβre told that the length of π΅πΆ is 55 centimeters. So thatβs the length of the side connecting vertex π΅ to vertex πΆ. Another way of describing this side is using a lowercase letter π because this side is opposite angle π΄. In the same way, we can describe the side π΄πΆ, which is opposite angle π΅, as lowercase π and the side π΄π΅, which is opposite angle πΆ, as lowercase π. Weβll describe the sides using these single letters π, π, and π in the remainder of this question.

The next piece of information weβre given is that π΄πΆ minus π΄π΅ is 13 centimeters. We can therefore form an equation. π΄πΆ, which weβre referring to as lowercase π, minus π΄π΅, which we are referring to as lowercase π, is equal to 13. The final piece of information weβre given is that the perimeter of the triangle is 124 centimeters. We can therefore form a second equation. The perimeter of a triangle is found by summing its three side lengths. So we have the equation π plus π plus π equals 124. But of course the value of π is 55. So we can substitute this value into our equation. We have 55 plus π plus π equals 124. And then subtracting 55 from each side of this equation gives a simplified equation. π plus π equals 69.

What we now have is a pair of linear simultaneous equations in the two variables π and π, which represent the unknown side lengths we wish to calculate. In order to find their values, we need to solve this pair of simultaneous equations. Thereβs more than one way we can do this. But if we observe that in the first equation we have negative π and in the second equation we have positive π, we should realize that we can eliminate this variable by adding the two equations together, because negative π plus π gives zero.

Adding the two equations then, we have π plus π, which is two π; negative π plus π, which cancel each other out to give zero; and on the right-hand side 13 plus 69, which is equal to 82. So we have an equation in π only. To solve this equation for π, we can divide both sides by two and we find that π is equal to 41. So we found the value of one unknown.

To find the value of the other, we need to substitute this value of π that weβve just calculated into either of our two equations. Letβs choose equation two. When we do this, we obtain 41 plus π equals 69. We can solve this equation for π by subtracting 41 from each side. And we find that π is equal to 28.

We should check our answer though by substituting the values of π and π that weβve calculated into equation one. The left-hand side of equation one is π minus π. So thatβs 41 minus 28, which is equal to 13. And as this is the same as the value on the right-hand side of equation one, this confirms that our solution is correct. Remember that π and π represent the lengths of sides π΄πΆ and π΄π΅, respectively. So we need to include the units, which are centimeters, with our answer.

So, by forming a pair of linear simultaneous equations and then solving them using the elimination method, weβve found that π΄πΆ is 41 centimeters and π΄π΅ is 28 centimeters. We were asked to give our answers to the nearest centimeter. But as the values we found were exact integer values anyway, there was no need to round our answers.