Respuesta :

LammettHash

Let [tex]\varphi[/tex] be the measure of the angle labeled in the attached image below. By the law of sines,

[tex]\dfrac{\sin\theta}{10x+5}=\dfrac{\sin(\pi-\varphi)}{78}=\dfrac{\sin\varphi}{78}\implies\sin\varphi=\dfrac{78\sin\theta}{10x+5}[/tex]

and

[tex]\dfrac{\sin\theta}{6x-1}=\dfrac{\sin\varphi}{42}\implies\sin\varphi=\dfrac{42\sin\theta}{6x-1}[/tex]

Then

[tex]\dfrac{78\sin\theta}{10x+5}=\dfrac{42\sin\theta}{6x-1}\implies78(6x-1)=42(10x+5)[/tex]

From here we can solve for [tex]x[/tex]:

[tex]468x-78=420x+210\implies48x=288\implies\boxed{x=6}[/tex].

Ver imagen LammettHash

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 [tex]\theta[/tex] 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:

[tex]\frac{42}{78} = \frac{6x - 1}{10x + 5}[/tex]

[tex]42(10x + 5) = 78(6x - 1)[/tex]

[tex]420x + 210 = 468x - 78[/tex]

[tex]48x = 288[/tex]

[tex]x = \frac{288}{48}[/tex]

[tex]x = 6[/tex]

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

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