Question Video: Finding the Length of a Side in a Triangle given the Corresponding Side's Length in a Congruent Triangle

In the given figure, 𝑆𝑃 = 3π‘₯ βˆ’ 7 and 𝑆𝑀 = 2π‘₯ βˆ’ 2. Find 𝑆𝑃.

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Video Transcript

In the given figure, 𝑆𝑃 equals three π‘₯ minus seven and 𝑆𝑀 equals two π‘₯ minus two. Find 𝑆𝑃.

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

Let’s start by noting that we have two right angles, angle 𝐽𝑀𝑆 and angle 𝐽𝑃𝑆. So, we have a pair of congruent angles. We can see that angle 𝑀𝐽𝑆 and angle 𝑃𝐽𝑆 from the diagram are also congruent. We’re given the lengths for 𝑆𝑃 and 𝑆𝑀, but we’re not told that these are congruent. And we’re not given any information about the line 𝐽𝑀 or 𝐽𝑃. We can see, however, that the line 𝐽𝑆 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 𝐽𝑀𝑆 and triangle 𝐽𝑃𝑆 are congruent using the angle-angle-side or AAS rule. Let’s see if this will help us to find the length 𝑆𝑃. Using the fact that these triangles are congruent, we know that the length 𝑆𝑃 in triangle 𝐽𝑃𝑆 is congruent with the length 𝑆𝑀 in triangle 𝐽𝑀𝑆. And so, two π‘₯ minus two must be equal to three π‘₯ minus seven. So, we’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, we’ll have three π‘₯ minus seven equals two π‘₯ minus two. Subtracting two π‘₯ from both sides of the equation, we’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 we’ve found the answer. However, we weren’t asked for π‘₯. We were asked for the length 𝑆𝑃.

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

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