Lesson video

Hi, I'm Mrs Dennett.

In this lesson, we're going to be using trigonometry to find the perpendicular height of a triangle.

In order for us to start calculating the perpendicular height of a triangle, you need to be able to identify it on a diagram.

Perpendicular means that we have two lines meeting each other at a right angle.

The height will split a right angle to the base.

In this question, we're told that the base is length b.

So y must be the perpendicular height, as it is perpendicular to the base, that's at right angles to the base and touches the top, the apex of the triangle formed by the other two sides.

Now we can start to calculate the perpendicular height of a triangle.

This is the perpendicular height of this triangle.

We label it with a lowercase h.

You can see, we have formed two right angle triangles, either side of the line, h.

We are given a side length and an angle for the triangle on the left.

So I can redraw this triangle and use trigonometry to find the heights.

We now label the sides with capital H for the hypotenuse, capital O for the opposite and a capital A for the adjacent.

Take care when labelling to use capital letters for these sides.

So as not to confuse the hypotenuse, which is a capital H, with the heights, which is a little h and is the opposite side.

So in this triangle, we want to find the length of the opposite side, our height and we are given the length of the hypotenuse.

So we can use the sine ratio to work out the height.

We solve this equation using our calculators.

So I multiply 19.

5 by sine 30, and we get the height, which is equal to 9.

75 centimetres.

Here's a different triangle.

It is isosceles.

I know this because of the small lines or hatch marks on the sides, which aren't labelled with a length.

For isosceles triangles, the perpendicular height is right in the centre of the base.

Again, we've got two right angle triangles here, we formed two right angle triangles, either side of the line h.

So we can use either triangle here as both angles adjacent to the base at 28 degrees.

But we'll use the left triangle as it's labelled with the angle already.

Now, with this triangle, we're only using half the base.

Remember the perpendicular height line, split the triangle, the larger triangle in half.

So we need to half the base to get 21 millimetres.

We then label with H, O and A.

Hypotenuse, opposite and adjacent.

So we can use trigonometry to find the perpendicular height.

So we can see in this triangle, we're going to be using the opposite and the adjacent sides, O an A.

So this indicates we have to substitute into the tangent ratio.

So we get tan 28 equals the height divided by 21.

We solve this equation using our calculators to get the height, which is 11.

17 millimetres to two decimal places.

Remember the units, they are in millimetres.

So we've got 11.

17 millimetres for the perpendicular height.

Here are some questions for you to try.

Pause the video to complete the task and restart when you were finished.

Here are the answers.

These are all isosceles triangles.

So you need to use the hatch marks, which show the equivalent side lines to help you in part A.

For part B, the base length needs to be halfed to six centimetres when using trigonometry to find the height.

In the last question, part C it is instead the angle, which needs to be halved to 35 degrees before applying trigonometry to find the perpendicular height.

Here's some questions for you to try, pause the video to complete the tasks and restart when you are finished.

Here are the answers.

For part A, you need to use the sine ratio to find the height and for part B, you needed to use the tan ratio.

We now want to work out the area of the triangle.

Can you remember how to do this? Here's a reminder.

We need to find the base length and the perpendicular height.

We use trigonometry to work out the height.

In a previous example, it was 9.

75 centimetres.

The base length is 17 plus four, which is 21 centimetres in total.

We now substitute into the formula and work out the area.

This is 102.

4 to one decimal place.

Remember to include the units, the area which are centimetre squared for this triangle.

Let's apply what we have just learned about area.

So this isosceles triangle from earlier too.

We want the area.

So we first need the perpendicular height.

Let's remind ourselves how we found this.

We halved the base and labelled with O and A because we didn't need the hypotenuse.

So we use the tan ratio to get the height, which was 11.

1658, et cetera.

Notice I'm not rounding my answer just yet as I want the area calculation to be as accurate as possible.

So we use 42 millimetres for the base.

Make sure you use the full base length here and we get 234.

5 centimetres when we round the area to one decimal place using our calculator.

Here's a question for you to try.

Pause the video to complete the task and restart when you are finished.

Here are the answers.

Remember to include the units for area in each of your answers for these questions.

If you find that your answer seems very close to the one given, check that you didn't round your answer too soon after finding the perpendicular height.

Here's a question for you to try.

Pause the video to complete the task and restart when you are finished.

Here's the answer, in this question, the triangle is equilateral.

So all of the sides are six centimetres.

The perpendicular height is in the centre of the base of the triangle.

So when using trigonometry to find the height, half the base, three centimetres.

All the angles will be 60 degrees as they're all equal because the triangle is equilateral.

You can use the sine or the tangent ratio to find the height of the triangle here.

Or you may have even spotted an opportunity to use Pythagoras theorem too.

The height is 5.

196152423, et cetera.

Or root 27, if you use Pythagoras.

So it's a little bit easier and potentially more accurate and a lot more accurate to use Pythagoras here.

Now, remember not to round the decimal.

If you got 5.

19, et cetera, until you have multiplied that decimal by a half times the base, or you will lose the accuracy for the area.

Only round to one decimal place at the very end.

That's all for this lesson.

Thank you for watching.