Lesson video

Hi, I'm Miss Davies.

In today's lesson, we're going to be applying Pythagoras' theorem to isosceles triangles.

With isosceles triangles, they have two equal sides and two equal angles.

This means we can split an isosceles triangle into two congruent triangles.

What type of triangles will these be.

We now have two congruent right angled triangles, both with a hypotenuse of 15 centimetres.

The original base was 18 centimetres.

What do you think the base of each right angled triangle is now? Well done If you said nine centimetres.

The original base has been divided into two equal parts.

In this example, we're trying to find the height of the isosceles triangle show.

We know that this length is 15 centimetres, As is an isosceles triangle.

An isosceles triangle have two equal sides.

We can split our isosceles triangle into two congruent right angled triangles, each with a base of nine centimetres.

We can then apply Pythagoras' theorem to calculate the height of the isosceles triangle.

Pythagoras' theorem states that A squared plus B squared equals C squared.

I am labelling our missing side which is the height as A.

We can now substitute these values into Pythagoras' theorem.

To get A squared at nine squared equals 15 squared.

This is the same as A squared at 81 is equal to 225.

We can rewrite this as A squared equals 225, subtract 81, or A squared equals 144.

To find the height of the triangle, we're going to calculate the square root of 144, which is 12.

The height of this isosceles triangle is 12 centimetres.

As this is an isosceles triangle, we know that this is 21 metres, as it shows that these two sides are equal.

In order to find the base of the isosceles triangle, we need to know the height of it.

What is the height of this triangle? It's 17 metres.

We now have two right angled triangles with a height of 17 metres and a hypotenuse of 21 metres.

To find the base, we're going to start by finding length B.

We're going to substitute these values into Pythagoras' theorem.

We can work out that length B is 12.

3 metres.

As length B is half of the base, we need to double this to find the whole base.

This gives us 24.

6 metres.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

In Part A and B, you need to half the base before you could apply Pythagoras' theorem.

This means that one of the shorter lengths in Part A, is five centimetres, and in part B, is six centimetres.

In Part C, you need to double the length that you found using Pythagoras' theorem to give the total base.

In Part D, we've got an equilateral triangle.

So this means that the hypotenuse of our rights angle triangle is going to be 15, and the base that we're going to use to apply Pythagoras' theorem to, is 7.

5.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here is the answer.

Leanne needs to double 9.

8 to get the whole of the base.

Here's a question for you to try.

Pause the video to complete your task and resume once you're finished.

To find the area of a triangle, we need to multiply the base by the height and divide by two.

To find the area of this triangle, we need to calculate the perpendicular height.

Through Pythagoras' theorem, I'm going to work out the height, which I've labelled as B.

I have found that the height of this triangle is 13.

74 centimetres.

To find the area of the triangle, I'm going to calculate base multiplied by height and divide it by two.

This is 12 multiplied by 13.

74 divided by two.

However, I'm going to use the square root of 189, instead of 13.

74, as this is more accurate.

This gives us 164.

97 divided by two, which is 82.

5 centimetres squared.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

In Part A, the perpendicular height is 36.

37 centimetres.

This gives an area of 963.

8 centimetres squared.

In Part B, the perpendicular height is 9.

05 millimetres, and the area is 209.

6 millimetres squared.

A regular pentagon can be split into five congruent equilateral triangles.

All sides have a length of eight centimetres.

We can then apply Pythagoras' theorem to this right angled triangle with a base of four centimetres to find the height.

The area of each triangle is found by multiplying eight by the square root of 48 and dividing by two.

This gives 27.

7 centimetres squared, as the area of each triangle.

This means the area of the whole Pentagon is found by multiplying 27.

7 by five.

This is 138.

56 centimetres squared.

Here are some questions for you to try.

Pause the video to complete your task and resume once you're finished.

Here are the answers.

In question four, the area of the rectangle is 18 metres squared.

The perpendicular height of the triangle is four metres.

The base we know is six metres.

This means the area of the triangle is 12 metres squared.

The total area of the compound shape is 30 metres squared.

In question five, the hexagon can be split into six Congruent equilateral triangles.

Each triangle has the area of 21.

22 centimetres squared.

This means the total area of the hexagon is 127.

3 centimetres squared.

That's all for this lesson.

Thanks for watching.