Two triangles are shown below. 𝐵 is a point on the straight line 𝐴𝐶. 𝐶𝐵 is equal to 𝐶𝐷. The angle 𝐵𝐶𝐷 is equal to 32 degrees. 𝐴𝐵 is equal to 2.3 centimetres. And 𝐵𝐸 is equal to 6.3 centimetres. Work out the length marked 𝑥.
Let’s look closely at the triangle 𝐴𝐵𝐸 for which we’re trying to find the length of one of its sides. It’s a non-right-angled triangle, and we already know the length of two of its sides. We’ll need to use non-right-angle trigonometry to find the other side. That’s the sine rule or the cosine rule. But we can’t do this without finding at least one of the angles in this triangle.
Let’s look at triangle 𝐵𝐶𝐷 then. 𝐶𝐷 and 𝐶𝐵 are equal in length. So this is an isosceles triangle. This means its base angles, these two, must be the same. Angles in a triangle sum to 180 degrees. So we subtract 32 from 180. And that gives us the total sum of the angles 𝐶𝐵𝐷 and 𝐶𝐷𝐵. 180 minus 32 is 148.
These angles are the same size, so we can halve 148 degrees to find each of the angles. 148 divided by two is 74. And we found the angles at 𝐶𝐵𝐷 and 𝐶𝐷𝐵 to be 74 degrees. This helps us because we can now find the angle 𝐴𝐵𝐸 shown. Angles on a straight line sum to 180 degrees. So we can find angle 𝐴𝐵𝐸 by subtracting 74 degrees from 180. That’s 106 degrees.
And we now have everything we need to find the length 𝑥. We said we’d need to use either the sine rule or the cosine rule. Let’s recall these. The sine rule is 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵 which equals 𝑐 over sin 𝐶. This is the form we generally use to find a missing length. We could use sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏 equals sin 𝐶 over 𝑐, but that’s preferable when you’re trying to find a missing angle. Either of these will work in either scenario. It’s just about reducing the amount of rearranging we need to do.
The cosine rule is 𝑎 squared equals 𝑏 squared plus 𝑐 squared minus two 𝑏𝑐 cos 𝐴. Notice how the sine rule relies on having pairs of sides and angles. To find a missing side, we’d need to know two of the angles. That means we’re going to use the cosine rule here.
To be able to use the cosine rule, we need an angle sandwich. It’s an angle enclosed between two known sides. Here that’s exactly what we have. We have an angle of 106 degrees sat between a side of 2.3 centimetres and a side of 6.3 centimetres. We will need to change the letters in our triangle to match our formula.
The angle in our formula is capital 𝐴, so we’ll label the vertex in our triangle where the angle of 106 degrees sits capital 𝐴. The side directly opposite that is then lowercase 𝑎. And we can label our remaining two sides as 𝑏 and 𝑐 in either order.
We substitute all these values into our formula. And it gives us 𝑥 squared is equal to 6.3 squared plus 2.3 squared minus two multiplied by 6.3 multiplied by 2.3 multiplied by cos of 106 degrees, which is 52.9679.
To solve this equation for 𝑥, we need to do the opposite of squaring, which is to square-root both sides of the equation. We’re going to use the exact value we just found on our calculator to prevent any issues from rounding too early. And we get the square root of 52.9679 and so on is 7.2779.
Now the question doesn’t tell us a suitable level of accuracy to use. So the three significant figures is generally a good choice. The first significant figure is the first nonzero digit. Here that’s the seven. This means two is the second significant figure, and the next seven is the third significant figure. We look to the digit immediately to the right of this. That’s called the deciding digit. Since the deciding digit is above five, we round our number up. That takes it to 7.28. And 𝑥 is therefore 7.28 centimetres.