$\begingroup$

I was studying for some exams when a wild question appeared. It looks like this:

Given that the perimeter of a triangle is $180$ inches. If the angles of the triangles are in the ratio $5:6:7$, determine the sides of the triangle.

My work:

Well, the only way that the angles found in the triangle to be in the ratio $5:6:7$ is when the three angles in the triangle were $50$, $60$ and $70$ degrees. I don't know what might be the three angles of the triangle if the ratio of those three becomes arbitrary, say 11:12:5.

I imagine holding a wire with length 180 inches. but I don't know how far from the tip of the wire will I bend to make an angle of 50 degrees and the succeeding angles in triangle.

I'm in a mess. How will you find the sides of the triangle?

$\endgroup$ 1

1 Answer

$\begingroup$

By law of sines:$$\frac{a}{\sin50^{\circ}}=\frac{b}{\sin60^{\circ}}=\frac{c}{\sin70^{\circ}}$$and $a+b+c=180$.

Let $\frac{a}{\sin50^{\circ}}=k$.

Hence, $a=k\sin50^{\circ}$, $b=k\sin60^{\circ}$, $c=k\sin70^{\circ}$ and$$k\sin50^{\circ}+k\sin60^{\circ}+k\sin70^{\circ}=180,$$which gives $$k=\frac{180}{\sin50^{\circ}+\sin60^{\circ}+\sin70^{\circ}}$$and from here we can now get the values of $a$, $b$ and $c$.

$\endgroup$ 2

Your Answer

Sign up or log in

Sign up using Google Sign up using Facebook Sign up using Email and Password Submit

Post as a guest

Post Your Answer Discard

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy