Question

Ifarighttriangle hastwolegs oflength9and12,andahypotenuse oflength15,findthelength of the two legs of a similar triangle with hypotenuse 25

Answer 1

Since we want to specify the length of the sides of a triangle from the known lengths of another triangle that is similar to the first one, we can express the ratios of their sides like this:

`(hypotenuse2)/(hypotenuse1)=(a2)/(a1)=(b2)/(b1)`

Where 1 means triangle 1, 2 means triangle 2, a is the first leg and b is the other leg.

Replacing the values that we know from the triangle one we get:

`\begin{gathered} (hypotenuse2)/(hypotenuse1)=(a2)/(a1)=(b2)/(b1) \\ (25)/(15)=(a2)/(9)=(b2)/(12) \end{gathered}`

Now, let's find the length of the second triangle with the first two ratios, like this:

`\begin{gathered} (25)/(15)=(a2)/(9) \\ (25)/(15)*9=(a2)/(9)*9 \\ (25)/(15)*9=a2 \\ a2=(25)/(15)*9=(5)/(3)*9=15 \end{gathered}`

And for the second length of the second triangle we can use the first and the last ratio, like this:

`\begin{gathered} (25)/(15)=(b2)/(12) \\ (25)/(15)*12=(b2)/(12)*12 \\ b2=(25)/(15)*12=(5)/(3)*12=20 \end{gathered}`

Then the second triangle has two legs of length 15 and 20