If 𝑐, 𝑏, and 𝑎 are three proportional side lengths in a triangle, where 𝑐 plus 𝑏 is equal to 18 and 𝑏 plus 𝑎 is equal to 10, find the ratio 𝑐 to 𝑏.
The fact that 𝑐, 𝑏, and 𝑎 are three proportional side lengths in a triangle leads us to an important rule or fact that we might or might not already know. This states that the side length 𝑐 divided by the side length 𝑏 is equal to the side length 𝑏 divided by the side length 𝑎. The first side divided by the second side is equal to the second side divided by the third side.
We are told in this question that the sum of side lengths 𝑐 and 𝑏 is equal to 18. If we subtract 𝑏 from both sides of this equation, we get 𝑐 is equal to 18 minus 𝑏. We are also told that 𝑏 plus 𝑎 is equal to 10. Once again, we can subtract 𝑏 from both sides of this equation. This gives us 𝑎 is equal to 10 minus 𝑏.
We can now substitute these expressions for 𝑐 and 𝑎 into the general rule. This gives us 18 minus 𝑏 divided by 𝑏 is equal to 𝑏 divided by 10 minus 𝑏. We can simplify this equation by cross multiplying, giving us 18 minus 𝑏 multiplied by 10 minus 𝑏 is equal to 𝑏 squared. Distributing the parentheses by using the FOIL method gives us 180 minus 18𝑏 minus 10𝑏 plus 𝑏 squared. Subtracting 𝑏 squared and adding 28𝑏 to both sides gives us 180 is equal to 28𝑏. Dividing both sides of this equation by 28 gives us 𝑏 is equal to 180 over 28. By dividing the numerator and denominator by four, this simplifies to 45 over seven. Our value of 𝑏 is 45 over seven or forty-five sevenths.
We can now substitute this back into the equation 𝑐 is equal to 18 minus 𝑏 to calculate the value of 𝑐. 18 multiplied by seven is 126. So 18 is one hundred and twenty-six sevenths. Subtracting forty-five sevenths from this gives us eighty-one sevenths. We were asked to find the ratio 𝑐 to 𝑏, so we will clear some space to do this. Our ratio gives us eighty-one sevenths to forty-five sevenths. We can multiply both sides of this ratio by seven, giving us 81 to 45. Finally, as these are both divisible by nine, this gives us nine to five. The ratio of 𝑐 to 𝑏 in its simplest form is nine to five.