Video: Using the Pythagorean Theorem to Determine If a Triangle is a Right Triangle

A triangle has vertices of the points ๐ด(4, 1), ๐ต(6, 2), and ๐ถ(2, 5). Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form. Is this triangle a right triangle?

03:55

Video Transcript

A triangle has vertices of the points ๐ด four, one; ๐ต six, two; and ๐ถ two, five. Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form. And secondly, is this triangle a right triangle?

Letโ€™s begin by sketching this triangle on a coordinate grid. We absolutely donโ€™t need to plot this triangle accurately. We arenโ€™t going to be measuring the lengths of any of the lines. We just want to sketch it using the approximate position of these three points relative to one another.

So the triangle looks a little something like this. Now, from our sketch, it looks possible that this could be a right triangle with the right angle at ๐ด. But we canโ€™t confirm this from our sketch. Letโ€™s consider the first part of the question. We need to find the lengths of the three sides of the triangle. And weโ€™ll begin by finding the length of the side ๐ด๐ต.

We can sketch in a right triangle below this line using ๐ด๐ต as its hypotenuse. We can also work out the lengths of the other two sides in this triangle. The horizontal side will be the difference between the ๐‘ฅ-values at its endpoints. Thatโ€™s the difference between six and four, which is two. And the vertical side will be the difference between the ๐‘ฆ-values at its endpoints. Thatโ€™s the difference between two and one, which is one.

As we now have the lengths of two sides in a right triangle and we wish to calculate the length of the third side, we can apply the Pythagorean theorem, which tells us that, in a right triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Remember, ๐ด๐ต is the hypotenuse. So we have that ๐ด๐ต squared is equal to one squared plus two squared. One squared is one and two squared is four. So adding these values together, we have that ๐ด๐ต squared is equal to five.

To find the length of ๐ด๐ต, we need to square root each side of this equation. And remember at this point, weโ€™ve been told to give our answer as a surd. So we have that ๐ด๐ต is equal to root five. We can find the lengths of the other two sides of the triangle in the same way. We sketch in a right triangle below the line ๐ต๐ถ. And we see that it has a horizontal side of four units and a vertical side of three units.

๐ต๐ถ is the hypotenuse of this triangle. So applying the Pythagorean theorem, we have that ๐ต๐ถ squared is equal to three squared plus four squared. Thatโ€™s nine plus 16, which is equal to 25. ๐ต๐ถ is therefore equal to the square root of 25, which is simply the integer five. In the same way, ๐ด๐ถ is the hypotenuse of a right triangle with shorter sides of two and four units. So ๐ด๐ถ is equal to the square root of 20, which simplifies to two root five.

So weโ€™ve answered the first part of the question. And now we need to determine whether this triangle is a right triangle. Well, if it is, then the Pythagorean theorem will hold for its three side lengths. Now we suspect it that the right angle was at ๐ด, which would make ๐ต๐ถ the hypotenuse of the triangle if it is indeed a right triangle.

We therefore want to know whether ๐ต๐ถ squared is equal to ๐ด๐ต squared plus ๐ด๐ถ squared. Well, we can in fact use the squared side lengths. We know that ๐ต๐ถ squared is 25. We know that ๐ด๐ต squared is five. And we know that ๐ด๐ถ squared is 20. So is it true that 25 is equal to five plus 20? Yes, of course, itโ€™s true, which means that the Pythagorean theorem holds for this triangle. And therefore, it is indeed a right triangle. So weโ€™ve completed the problem. We have the three side lengths. ๐ด๐ต equals root five, ๐ต๐ถ equals five, and ๐ด๐ถ equals two root five. And weโ€™ve determined that the triangle is a right triangle.

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