Video: Using the Cosine Rule to Find the Side Lengths of an Isosceles Triangle

Find the length of side ๐‘Ž of the triangle shown.

02:24

Video Transcript

Find the length of side ๐‘Ž of the triangle shown.

In our triangle, we see this side length opposite an angle of 103 degrees. Along with this, based on these markings on the other two sides of the triangle, we can tell that they have the same length. So this side length here is 7.1 centimeters also. As a side note, this means weโ€™re working with a special kind of triangle called an isosceles triangle. But anyway, our mission is to solve for this side length here, and we can do it using a rule called the cosine rule, also sometimes called the law of cosines.

This rule tells us that given a triangle with interior angles marked out capital ๐ด, ๐ต, and ๐ถ and corresponding side lengths lowercase ๐‘Ž and ๐‘ and ๐‘, we can solve for the square of one of the sides by adding together the square of the other two sides and then subtracting from that two times those other two sides multiplied by the cos of an angle weโ€™ve called capital ๐ด, where in our triangle, this angle capital ๐ด is opposite the side length lowercase ๐‘Ž that weโ€™re solving for. So in our actual triangle over here, to solve for this side length, weโ€™ll apply the cosine rule.

To do this, weโ€™ll need to identify what the other two side lengths, theyโ€™re called ๐‘ and ๐‘ in this equation, and the angle opposite the side length we want to solve for, called capital ๐ด here, are. Looking at our triangle, we see that angle opposite the side length weโ€™re solving for is 103 degrees. So thatโ€™s our angle capital ๐ด. And then, as far as identifying ๐‘ and ๐‘, these values in our cosine rule equation, we see that the other two side lengths in our triangle are both the same. In this case then, both ๐‘ and ๐‘ are 7.1 centimeters. This is the case because, as we saw earlier, weโ€™re working with an isosceles triangle.

Knowing all this, we can now plug in these values into their place on the right-hand side of our cosine rule expression. And once weโ€™ve done this, weโ€™re very close to being able to solve for the unknown side length ๐‘Ž. The one thing we donโ€™t want to forget, though, is that right now we have an expression for ๐‘Ž squared. To solve just for ๐‘Ž, weโ€™ll want to take the square root of both sides, where on the left, this square root and the square term will cancel one another out. And now, when we enter this expression on the right-hand side into a calculator, rounding to two significant figures, we find a result of 11 centimeters. This is the length of side ๐‘Ž in our triangle.

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