Question

Please Help Find the value of x

Answer 1

Let $\varphi$ be the measure of the angle labeled in the attached image below. By the law of sines,

$\dfrac{\sin\theta}{10x+5}=\dfrac{\sin(\pi-\varphi)}{78}=\dfrac{\sin\varphi}{78}\implies\sin\varphi=\dfrac{78\sin\theta}{10x+5}$

and

$\dfrac{\sin\theta}{6x-1}=\dfrac{\sin\varphi}{42}\implies\sin\varphi=\dfrac{42\sin\theta}{6x-1}$

Then

$\dfrac{78\sin\theta}{10x+5}=\dfrac{42\sin\theta}{6x-1}\implies78(6x-1)=42(10x+5)$

From here we can solve for $x$:

$468x-78=420x+210\implies48x=288\implies\boxed{x=6}$.

Answer 2

Using similarity of triangles, it is found that x = 6.

What are similar triangles?

Similar triangles have the same angles, and the length of their equivalent sides is proportional.

In this problem:

• The bisection at angle $\theta$ forms two similar triangles.

• Side of 42 is equivalent to the side of 78, and the side of 6x - 1 is equivalent to the side of 10x + 5.

Hence, applying the ratios:

$\frac{42}{78} = \frac{6x - 1}{10x + 5}$

$42(10x + 5) = 78(6x - 1)$

$420x + 210 = 468x - 78$

$48x = 288$

$x = \frac{288}{48}$

$x = 6$

To learn more about similarity of triangles, you can take a look at https://brainly.com/question/11899908