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# Question Video: Solving a System of Linear Equations to Find Unknown Side Lengths in a Triangle Involving the Perimeter Mathematics • 8th Grade

π΄π΅πΆ is a triangle, where π΅πΆ = 55 cm, π΄πΆ β π΄π΅ = 13 cm, and the perimeter is 124 cm. Find the lengths of the line segments π΄πΆ and π΄π΅, giving the answers to the nearest centimeter.

04:29

### Video Transcript

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

Letβs begin by thinking about the notation used to describe this triangle. The triangle has been described as π΄π΅πΆ. So these are the letters corresponding to its three vertices. Weβre told that the length of π΅πΆ is 55 centimeters. So thatβs the length of the side connecting vertex π΅ to vertex πΆ. 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΄πΆ, which is opposite angle π΅, as lowercase π and the side π΄π΅, which is opposite angle πΆ, as lowercase π. Weβll describe the sides using these single letters π, π, and π in the remainder of this question.

The next piece of information weβre given is that π΄πΆ minus π΄π΅ is 13 centimeters. We can therefore form an equation. π΄πΆ, which weβre 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 weβre 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. Thereβs 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 weβve just calculated into either of our two equations. Letβs 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 weβve calculated into equation one. The left-hand side of equation one is π minus π. So thatβs 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΄πΆ 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, weβve found that π΄πΆ 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.

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