Video: CBSE Class X • Pack 3 • 2016 • Question 8

CBSE Class X • Pack 3 • 2016 • Question 8

04:37

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.

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