### Video Transcript

Prove that the points three, zero;
six, four; and negative one, three are the vertices of a right-angled isosceles
triangle.

To show that the triangle is
right-angled, we need to prove that the lengths of the three sides of the triangle
satisfy the Pythagorean theorem. Remember, this says that the square
of the longest side is equal to the sum of the squares of the smaller two sides. It’s often written as 𝑎 squared
plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the length of the hypotenuse in the
triangle.

We also need to show that the
triangle is isosceles. That is to say, we need to show
that the lengths of two of the sides of the triangle are equal. We can use the distance formula to
help us calculate the lengths of the sides of this triangle. The distance formula is derived
from the Pythagorean theorem. And it says that for two points 𝑥
one, 𝑦 one and 𝑥 two, 𝑦 two, the distance between them is given by the square
root of the difference between the 𝑥-coordinates squared plus the difference
between the 𝑦-coordinates squared.

Let’s begin by calculating the
distance between the points three, zero and six, four. This will tell us the length of
this side of the triangle. The difference between the
𝑥-coordinates is given by six minus three. And the difference between the
𝑦-coordinates is four minus zero. Evaluating each of these, and we
can see that the distance between these two points is given by the square root of
three squared plus four squared.

Now, you might have noticed that
these two numbers form part of a Pythagorean triple. Let’s complete the calculation
before looking at what that means. Three squared plus four squared is
nine plus 16, which is 25. And the square root of 25 is
five.

Remember, we said that three and
four are two numbers from a Pythagorean triple. A Pythagorean triple is a series of
three integers for which the sum of the squares of the two smaller integers is equal
to the square of the larger integer. In this case, three squared plus
four squared equals five squared. So we could’ve saved ourself a
little bit of time and spotted that the distance between these two points was
five. In fact, since these are points
plotted on a coordinate grid, we can say that the distance between them is five
units. Let’s repeat this process for the
point three, zero and negative one, three.

This time, it’s the square root of
negative one minus three squared plus three minus zero squared, which simplifies to
the square root of negative four squared plus three squared. We know that’s simply five. So the distance between three, zero
and negative one, three is five units. We have now shown that the lengths
of two sides of this triangle are equal. It’s definitely an isosceles
triangle. All we need to do now is prove the
right angle part. Let’s find the distance between the
last two points.

This time, the difference between
the 𝑥-coordinates is given by negative one minus six. And the difference between the
𝑦-coordinates is three minus four. That simplifies to negative seven
squared and negative one squared. Negative seven squared is 49. And negative one squared is
one. When we find the sum of these, we
get 50. So the distance between these two
points is the square root of 50 units.

We now have the lengths of the
three sides of our triangle. Let’s now substitute them into our
formula for the Pythagorean theorem. Remember, 𝑐 is the hypotenuse. It’s the longest side of our
triangle. We know that seven squared is
49. So the square root of 50 is likely
to be a little bit bigger than seven. Seven is larger than five. So the side of the triangle that
measures root 50 units is the hypotenuse. 𝑎 squared plus 𝑏 squared becomes
five squared plus five squared. Five squared is 25. So that in turn becomes 25 plus 25,
which is 50. 𝑐 squared becomes the square root
of 50 squared, which is once again 50.

We have now proved that in our
formula, 𝑎 squared plus 𝑏 squared equals 𝑐 squared. The sum of the squares of the
smaller two sides is equal to the square of the larger side. Since the sides of our triangle
satisfy the formula for the Pythagorean theorem, it must be right angled. And since two of the sides are of
equal length, it’s also isosceles.