The diameter of a circle 𝐴𝐷 is 82 centimetres. 𝐴𝐵 and 𝐴𝐶 are two chords on opposite sides of the circle, with lengths 5.1
centimetres and 48.4 centimetres, respectively. Find the length 𝐵𝐶, giving the answer to two decimal places.
It’s always sensible to begin by sketching out a diagram. This doesn’t need to be to scale, but it should be roughly in proportion so we can
check the suitability of any answers we get. This does look a bit tricky to start, but there are some circle theorems we can use
that will make things easier.
First, let’s add the chords 𝐵𝐷 and 𝐶𝐷. Remember, the angle subtended by the diameter is always 90 degrees. That means then that angle 𝐴𝐵𝐷 and 𝐴𝐶𝐷 are both right angles. We have two right-angled triangles, so we can use right angle trigonometry to
calculate the measure of angle 𝐴𝐷𝐵 and angle 𝐴𝐷𝐶. Let’s start with triangle 𝐴𝐵𝐷.
Side 𝐴𝐷 is the hypotenuse of the triangle. It’s the longest side, and it can be found by looking directly opposite the right
angle. Side 𝐴𝐵 is the opposite. It’s the side opposite to the given angle 𝜃.
Since we know the length of the opposite and the hypotenuse, we can use the sine
ratio to calculate the size of the angle 𝜃. Substituting the relevant values in gives us sin 𝜃 is equal to 5.1 over 82. To calculate the value of 𝜃, we’ll find the inverse sine of both sides of the
equation. The inverse sin of 5.1 over 82 is 3.565, so 𝜃 is 3.565 degrees. We won’t round this number just yet. Instead, we’ll use its exact form in any future calculations.
Now let’s look at triangle 𝐴𝐷𝐶. Once again, we know the hypotenuse of this triangle and the length of its opposite
side. We can substitute these values into our formula for the sine ratio. Sin 𝜃 is equal to 48.4 over 82. Once again, we find the inverse sine of both sides of our equation. The inverse sine of 48.4 over 82 is 36.174.
Next, there are a couple of circle theorems we can now use. We know the opposite angles in a cyclic quadrilateral must sum to 180 degrees. We can therefore calculate the measure of the angle 𝐵𝐴𝐶 by subtracting the two
angles we’ve just worked out from 180 degrees. That gives us 140.259. We also know that angles in the same segment are equal. This means then that angle 𝐴𝐵𝐶 must be equal to angle 𝐴𝐷𝐶. It’s also 36.174 degrees.
Redrawing the triangle 𝐴𝐵𝐶, we can see we have a non-right-angled triangle for
which we know the measure of two angles and the length of one of the sides. We can use the law of sines to help us find the length of 𝐵𝐶. The law of sines says 𝑎 over sin 𝐴 is equal to 𝑏 over sin 𝐵, which is equal to 𝑐
over sin 𝐶. Alternatively, that’s sometimes written as sin 𝐴 over 𝑎 equals sin 𝐵 over 𝑏,
which equals sin 𝐶 over 𝑐.
Now the law of sines can be used in either form. Since we’re trying to find the length of a side though, we’ll use the first form. This will minimise the amount of rearranging we need to do. Similarly, if we’re trying to find the size of the angle, we would use the second
Since we know the measure of the angle at 𝐴 and we’re looking to find the length of
the side 𝑎 and we know the measure of the angle at 𝐵 and the length of the side 𝑏
— that’s 𝐴𝐶 — we use 𝑎 over sin 𝐴 equals 𝑏 over sin 𝐵.
Substituting in the exact values gives us 𝑎 over sine of 140.259 equals 48.4 over
sine of 36.174. We can solve this equation by multiplying both sides by sine of 140.259. That gives us that 𝑎 is equal to 48.4 over sine of 36.174 multiplied by sine of
140.259. And popping that into our calculator gives us a value of 52.423. Correct to two decimal places, the length 𝐵𝐶 is 52.42 centimetres.