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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

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