### Video Transcript

Two triangles are shown below. π΅ is a point on the straight line π΄πΆ. πΆπ΅ is equal to πΆπ·. The angle π΅πΆπ· is equal to 32 degrees. π΄π΅ is equal to 2.3 centimetres. And π΅πΈ is equal to 6.3 centimetres. Work out the length marked π₯.

Letβs look closely at the triangle π΄π΅πΈ for which weβre trying to find the length of one of its sides. Itβs a non-right-angled triangle, and we already know the length of two of its sides. Weβll need to use non-right-angle trigonometry to find the other side. Thatβs the sine rule or the cosine rule. But we canβt do this without finding at least one of the angles in this triangle.

Letβs look at triangle π΅πΆπ· then. πΆπ· and πΆπ΅ 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 πΆπ΅π· and πΆπ·π΅. 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 πΆπ΅π· and πΆπ·π΅ to be 74 degrees. This helps us because we can now find the angle π΄π΅πΈ shown. Angles on a straight line sum to 180 degrees. So we can find angle π΄π΅πΈ by subtracting 74 degrees from 180. Thatβs 106 degrees.

And we now have everything we need to find the length π₯. We said weβd need to use either the sine rule or the cosine rule. Letβs recall these. The sine rule is π over sin π΄ equals π over sin π΅ which equals π over sin πΆ. This is the form we generally use to find a missing length. We could use sin π΄ over π equals sin π΅ over π equals sin πΆ over π, but thatβs preferable when youβre trying to find a missing angle. Either of these will work in either scenario. Itβs just about reducing the amount of rearranging we need to do.

The cosine rule is π squared equals π squared plus π squared minus two ππ cos π΄. Notice how the sine rule relies on having pairs of sides and angles. To find a missing side, weβd need to know two of the angles. That means weβre going to use the cosine rule here.

To be able to use the cosine rule, we need an angle sandwich. Itβs an angle enclosed between two known sides. Here thatβs 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 π΄, so weβll label the vertex in our triangle where the angle of 106 degrees sits capital π΄. The side directly opposite that is then lowercase π. And we can label our remaining two sides as π and π in either order.

We substitute all these values into our formula. And it gives us π₯ 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 π₯, we need to do the opposite of squaring, which is to square-root both sides of the equation. Weβre 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 doesnβt 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 thatβs 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. Thatβs called the deciding digit. Since the deciding digit is above five, we round our number up. That takes it to 7.28. And π₯ is therefore 7.28 centimetres.