Chapter 10.1, Problem 28E

To determine

To find whether or not the angle measure and side lengths of a triangle can have the same ratio of $3:4:5$ and whose side lengths are in the same ratio.

Answer to Problem 28E

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.

Explanation of Solution

Given information:

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

Calculation:

According to the sine rule, in a triangle with angles A, B, C and side lengths a, b and c

  $\frac{a}{\mathrm{Sin}A}=\frac{b}{\mathrm{Sin}B}=\frac{c}{\mathrm{Sin}C}$

Above can be rearranged as

  $a:b=\mathrm{Sin}A:\mathrm{Sin}B$

And

  $b:c=\mathrm{Sin}B:\mathrm{Sin}C$

Hence ratio of the side lengths are 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 is $\mathrm{Sin}(45):\mathrm{Sin}(60):\mathrm{Sin}(75)$ which is not same as $3:4:5$

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.