𝐴𝐵𝐶 is a triangle, where 𝐵𝐶 equals 55 centimeters, 𝐴𝐶 minus 𝐴𝐵 equals 13 centimeters, and the perimeter is 124 centimeters. Find the lengths of the line segments 𝐴𝐶 and 𝐴𝐵, giving the answers to the nearest centimeter.
Let’s begin by thinking about the notation used to describe this triangle. The triangle has been described as 𝐴𝐵𝐶. So these are the letters corresponding to its three vertices. We’re told that the length of 𝐵𝐶 is 55 centimeters. So that’s the length of the side connecting vertex 𝐵 to vertex 𝐶. Another way of describing this side is using a lowercase letter 𝑎 because this side is opposite angle 𝐴. In the same way, we can describe the side 𝐴𝐶, which is opposite angle 𝐵, as lowercase 𝑏 and the side 𝐴𝐵, which is opposite angle 𝐶, as lowercase 𝑐. We’ll describe the sides using these single letters 𝑎, 𝑏, and 𝑐 in the remainder of this question.
The next piece of information we’re given is that 𝐴𝐶 minus 𝐴𝐵 is 13 centimeters. We can therefore form an equation. 𝐴𝐶, which we’re referring to as lowercase 𝑏, minus 𝐴𝐵, which we are referring to as lowercase 𝑐, is equal to 13. The final piece of information we’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 𝑎 plus 𝑏 plus 𝑐 equals 124. But of course the value of 𝑎 is 55. So we can substitute this value into our equation. We have 55 plus 𝑏 plus 𝑐 equals 124. And then subtracting 55 from each side of this equation gives a simplified equation. 𝑏 plus 𝑐 equals 69.
What we now have is a pair of linear simultaneous equations in the two variables 𝑏 and 𝑐, 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. There’s more than one way we can do this. But if we observe that in the first equation we have negative 𝑐 and in the second equation we have positive 𝑐, we should realize that we can eliminate this variable by adding the two equations together, because negative 𝑐 plus 𝑐 gives zero.
Adding the two equations then, we have 𝑏 plus 𝑏, which is two 𝑏; negative 𝑐 plus 𝑐, 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 𝑏 only. To solve this equation for 𝑏, we can divide both sides by two and we find that 𝑏 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 𝑏 that we’ve just calculated into either of our two equations. Let’s choose equation two. When we do this, we obtain 41 plus 𝑐 equals 69. We can solve this equation for 𝑐 by subtracting 41 from each side. And we find that 𝑐 is equal to 28.
We should check our answer though by substituting the values of 𝑏 and 𝑐 that we’ve calculated into equation one. The left-hand side of equation one is 𝑏 minus 𝑐. So that’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 𝑏 and 𝑐 represent the lengths of sides 𝐴𝐶 and 𝐴𝐵, 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, we’ve found that 𝐴𝐶 is 41 centimeters and 𝐴𝐵 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.