Respuesta :

Focus on the sub-triangle on the left. It is a right triangle with legs 9 and 6, so its hypothenuse is

[tex]\sqrt{9^2+6^2}=\sqrt{81+36}=\sqrt{117}[/tex]

Now focus on the sub-triangle on the right. It is a right triangle with legs 6 and x, so its hypothenuse is

[tex]\sqrt{6^2+x^2}=\sqrt{x^2+36}[/tex]

Now, the entire triangle has legs [tex]\sqrt{117}[/tex] and [tex]\sqrt{x^2+36}[/tex], and its hypothenuse is [tex]9+x[/tex]. Write the Pytagorean theorem one last time to get

[tex]117+(x^2+36)=(9+x)^2\iff x^2+153=81+18x+x^2 \iff 18x+81=153[/tex]

Subtract 81 from both sides to get

[tex]18x=72 \iff x=\dfrac{72}{18}=4[/tex]

Answer: x = 4

Step-by-step explanation:

The attached photo shows a clearer illustration of the given triangle.

Looking at the photo, assuming ∆BCD is a right angle triangle. To determine BC, we would apply Pythagoras theorem which is expressed as

Hypotenuse² = opposite side² + adjacent side²

BC² = 9² + 6²

BC² = 81 + 36 = 117

BC = √117

To determine θ, we would apply the tangent trigonometric ratio.

Tan θ opposite side/adjacent side

Tan θ = 6/9 = 0.6667

θ = 33.6914

Considering ∆ABC,

Hypotenuse = x + 9

Adjacent = √117

Cos θ = adjacent side/ hypotenuse

Cos 33.6914 = √117/(x + 9)

Cross multiplying, it becomes

0.8320 = √117/(x + 9)

x + 9 = √117/0.8320

x + 9 = 13

x = 13 - 9 = 4

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