Question Video: Finding the Length of a Side in a Triangle given the Corresponding Side's Length in a Congruent Triangle Mathematics • 8th Grade

In the given figure, ππ = 3π₯ β 7 and ππ = 2π₯ β 2. Find ππ.

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Video Transcript

In the given figure, ππ equals three π₯ minus seven and ππ equals two π₯ minus two. Find ππ.

We can begin by filling in the given information about the lengths onto the diagram. Weβre not told any information about this figure other than the fact that there is a pair of right angles and thereβs also another pair of congruent angles. So, letβs check if these two triangles are congruent.

Letβs start by noting that we have two right angles, angle π½ππ and angle π½ππ. So, we have a pair of congruent angles. We can see that angle ππ½π and angle ππ½π from the diagram are also congruent. Weβre given the lengths for ππ and ππ, but weβre not told that these are congruent. And weβre not given any information about the line π½π or π½π. We can see, however, that the line π½π is common to both triangles, which means that we have a pair of congruent sides.

As we found two angles and a nonincluded side, we could say that triangle π½ππ and triangle π½ππ are congruent using the angle-angle-side or AAS rule. Letβs see if this will help us to find the length ππ. Using the fact that these triangles are congruent, we know that the length ππ in triangle π½ππ is congruent with the length ππ in triangle π½ππ. And so, two π₯ minus two must be equal to three π₯ minus seven. So, weβll need to solve to find the value of π₯.

If we rewrite this so that we have our higher coefficient of π₯ on the left-hand side, weβll have three π₯ minus seven equals two π₯ minus two. Subtracting two π₯ from both sides of the equation, weβll have π₯ minus seven equals negative two. We can then add seven to both sides of the equation, so π₯ equals negative two plus seven. And therefore, π₯ is equal to five. It can be easy to stop at this point and think that weβve found the answer. However, we werenβt asked for π₯. We were asked for the length ππ.

We were given that ππ equals three π₯ minus seven. So, we plug in our value of π₯ equals five into this equation, which gives us ππ equals three times five minus seven. And as three times five is 15, weβll have 15 minus seven, which is eight. And so, we have our answer ππ equals eight, which we found by proving that the two triangles were congruent.