Question

One angle of a quadrilateral has measure $\frac{2\pi^{c}}{5}$ and the measures of other three angles are in the ratio 2:3:4. Find their measures in radians and in degrees.

Answer 1

Answer:

64° ,  96° , 128°

16π/45 , 8π/15 , 32π/45

Step-by-step explanation:

2π/5 = 2 * 180/5 = 72°

Let say other three angles are

2k , 3k & 4k

2k + 3k + 4k + 72° = 360°

=> 9k = 288°

=> k = 32°

2k = 64°

3k = 96°

4k = 128°

2π/5 + 2k + 3k + 4k = 2π

=> 9k = 2π - 2π/5

=> 9k = 8π/5

=> k = 8π/45

2k = 16π/45

3k = 24π/45 = 8π/15

4k = 32π/45

Answer 2

Answer:

Given : One angle of a quadrilateral has measure $\frac{2\pi^{c}}{5}$and the measures of other three angles are in the ratio 2:3:4.

To Find :  Find their measures in radians and in degrees.

Solution:

One angle of a quadrilateral has measure $\frac{2\pi^{c}}{5}$

Convert it in degrees

$1 rad = \frac{180^{\circ}}{\pi}$

So, $\frac{2\pi^{c}}{5}= \frac{180^{\circ}}{\pi} \times\frac{2\pi}{5}$

$\frac{2\pi^{c}}{5}= 180 \times\frac{2}{5}$

$\frac{2\pi^{c}}{5}= 72^{\circ}$

Now we are given that the measures of other three angles are in the ratio 2:3:4.

Let the ratio be x

So, the angles are 2x , 3x and 4x

Property : Sum of all angles of quadrilateral is 360°

So, $72+2x+3x+4x=360$

$72+9x=360$

$9x=288$

$x=32$

So, the angles are $2x = 2\times 32 = 64^{\circ}$ , $3x = 3\times 32 = 96^{\circ}$ and$4x = 4\times 32 = 128^{\circ}$

So, The angles in Degrees are 72°,64°,96° and 128°

Now convert these angles in radians

$1^{\circ}=\frac{\pi}{180^{\circ}}$

So, $72^{\circ}=\frac{\pi}{180^{\circ}} \times 72$

$72^{\circ}=\frac{2\pi}{5}$

$64^{\circ}=\frac{\pi}{180^{\circ}} \times 64$

$64^{\circ}=\frac{16\pi}{45}$

$96^{\circ}=\frac{\pi}{180^{\circ}} \times 96$

$96^{\circ}=\frac{8\pi}{15}$

$128^{\circ}=\frac{\pi}{180^{\circ}} \times 128$

$128^{\circ}=\frac{32\pi}{45}$

So, The angles in radians are $\frac{2\pi}{5}$,  $\frac{16\pi}{45}$, $\frac{8\pi}{15}$ and  $\frac{32\pi}{45}$