### Video Transcript

A triangle has vertices of the
points ๐ด four, one; ๐ต six, two; and ๐ถ 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?

Letโs begin by sketching this
triangle on a coordinate grid. We absolutely donโt need to plot
this triangle accurately. We arenโt 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 ๐ด. But we canโt confirm this from our
sketch. Letโs consider the first part of
the question. We need to find the lengths of the
three sides of the triangle. And weโll begin by finding the
length of the side ๐ด๐ต.

We can sketch in a right triangle
below this line using ๐ด๐ต 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 ๐ฅ-values at its endpoints. Thatโs the difference between six
and four, which is two. And the vertical side will be the
difference between the ๐ฆ-values at its endpoints. Thatโs 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, ๐ด๐ต is the
hypotenuse. So we have that ๐ด๐ต 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 ๐ด๐ต squared is equal to five.

To find the length of ๐ด๐ต, we need
to square root each side of this equation. And remember at this point, weโve
been told to give our answer as a surd. So we have that ๐ด๐ต 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 ๐ต๐ถ. And we see that it has a horizontal
side of four units and a vertical side of three units.

๐ต๐ถ is the hypotenuse of this
triangle. So applying the Pythagorean
theorem, we have that ๐ต๐ถ squared is equal to three squared plus four squared. Thatโs nine plus 16, which is equal
to 25. ๐ต๐ถ is therefore equal to the
square root of 25, which is simply the integer five. In the same way, ๐ด๐ถ is the
hypotenuse of a right triangle with shorter sides of two and four units. So ๐ด๐ถ is equal to the square root
of 20, which simplifies to two root five.

So weโve 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 ๐ด, which would make ๐ต๐ถ the hypotenuse of the triangle if it is
indeed a right triangle.

We therefore want to know whether
๐ต๐ถ squared is equal to ๐ด๐ต squared plus ๐ด๐ถ squared. Well, we can in fact use the
squared side lengths. We know that ๐ต๐ถ squared is
25. We know that ๐ด๐ต squared is
five. And we know that ๐ด๐ถ squared is
20. So is it true that 25 is equal to
five plus 20? Yes, of course, itโs true, which
means that the Pythagorean theorem holds for this triangle. And therefore, it is indeed a right
triangle. So weโve completed the problem. We have the three side lengths. ๐ด๐ต equals root five, ๐ต๐ถ equals
five, and ๐ด๐ถ equals two root five. And weโve determined that the
triangle is a right triangle.