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# Trigonometry (pls help)

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A square with an area of 9 is surrounded by four congruent right triangles, forming a larger square with an area of 89. Find $$\tan \theta.$$

Jul 11, 2022

#1
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$$\tan \theta = \dfrac{4}{3}$$

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Jul 11, 2022
#2
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$$\tan \theta = \frac{8}{5}$$

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Jul 11, 2022
#3
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Let the longer side of the triangle be $$x+3$$. This means that the shorter side of the triangle is length $$x$$.

Now, let the total area of the triangle be $$a$$. We have the equation $$4a + 9 = 89$$, meaning $$a = 20$$.

The area of the triangle is $$x (x+3) \div 2$$, but we know that the area of each triangle is 20, so we have the equation: $${x(x+3) \over 2} = 20$$, meaning $$x = 5$$.

This means that the side opposite to $$\theta$$ is 8, and the side adjacent to it is 5, so $$\tan \theta = {\text{opposite} \over \text{adjacent}} =\color{brown}\boxed {8 \over 5}$$

Jul 12, 2022