Question

Is it possible to have a triangle whose angle measures are in the ratio $3:4:5$and whose side lengths are in the same ratio? Explain.

Short answer

It is not possible to have a triangle whose angle measures are in the ratio $3:4:5$ and whose side lengths are in the same ratio.

Step-by-step solution

Step 1. Given Information.

Ratio of the angle measures in a triangle is given as$3:4:5$.

Step 2. Calculation.

According to the sine rule, in a triangle with angles A, B, C and side lengths a, b and c,$\frac{a}{SinA}=\frac{b}{SinB}=\frac{c}{SinC}$.

Above can be rearranged as,

$a:b=SinA:SinB$

And

$b:c=SinB:SinC$

Hence ratio of the side lengths is proportionate to the Sine of angles opposite to the side and hence cannot be same ratio as the angles.

For the given example: let us assume the angles are 3x, 4x and 5x.

$3x+4x+5x=180°$

Now, add all the x together.

$12x=180$

Now, divide both the sides with 12.

$\frac{12x}{12}=\frac{180°}{12}$

This gives the value of$x=15$.

First angle is 3x, multiply 3 with 15.

$3\times 15=45°$

Second angle is 4x, multiply 4 with 15.

$4\times 15=60°$

Third angle is 5x, multiply 5 with 15

$5\times 15=75°$

So, the 3 angles are 45, 60 and 75.

The side ratio isrole="math" localid="1649367543087" $\mathrm{Sin}\left(45\right):\mathrm{Sin}\left(60\right):\mathrm{Sin}\left(75\right)$ which is not same as$3:4:5$.

Step 3. Conclusion.

It is not possible to have a triangle whose angle measures are in the ratio $3:4:5$ and whose side lengths are in the same ratio.