𝐴𝐵𝐶 is a triangle, where two sin
𝐴 equals three sin 𝐵 equals four sin 𝐶 and the perimeter is 169 centimetres. Find the values of 𝑎 and 𝑐,
giving the answer to the nearest centimetre.
We know the relationships between
the sin of 𝐴, sin of 𝐵, and sin of 𝐶. Notice that this equation looks a
little bit like the law of sines. If we compare this to the formula
for the law of sines, we can see that the coefficient for sin 𝐴, sin 𝐵, and sin 𝐶
in the general form are all fractions.
In order to make our equation look
like this, we’ll need to divide through by some common multiple of two, three, and
four. The lowest common multiple of these
three numbers is 12. So let’s begin by dividing through
our entire equation by 12. And simplified, we get two sin 𝐴
over 12 equals three sin 𝐵 over 12, which equals four sin 𝐶 over 12.
Let’s simplify each of these
fractions. We can divide through this first
fraction by a common factor of two. That gives us sin 𝐴 over six. We can divide through the second
fraction by a common factor of three. That gives us sin 𝐵 over four. And finally, dividing through by a
common factor of four for the third fraction gives us sin 𝐶 over three.
sin 𝐴 over six is equal to sin 𝐵
over four, which is equal to sin 𝐶 over three. The denominators of these fractions
correspond to the length of 𝑎, 𝑏, and 𝑐 in our formula for the law of sines. This means then that the ratio of
the length of the sides of the triangle 𝑎 to 𝑏 to 𝑐 can be written as six to four
Now, we know that the perimeter of
the shape is 169 centimetres, where the perimeter is the sum total of the length of
the triangles. To calculate the individual length
of the triangle then, we’ll need to share 169 centimetres in the ratio of six to
four to three. Six plus four plus three is 13. There are a total of 13 parts.
𝑎 is worth six of these. 𝑎 must be six thirteenths of the
perimeter. That’s six thirteenths of 169. Six thirteenths of 169 is 78. So 𝑎 must be 78 centimetres. 𝑐 is worth three of those 13
parts. It’s three thirteenths of the
perimeter. Three thirteenths of 169 is 39. So 𝑐 must be 39 centimetres.
We can check our answer by
calculating the length of 𝑏, which is four thirteenths of the perimeter. Four thirteenths of 169 is 52. So 𝑏 is 52 centimetres. If we add these three lengths, we
get 169, which is the perimeter. This gives us a strong indication
that the calculations we’ve performed are correct.
𝑎 is 78 centimetres and 𝑐 is 39