Question

Using the figure below, find the value of x. Enter your answer as a simplified radical or improper fraction (if necessary)
into the equation box below.

Answer 1

Answer:

x = 5$\sqrt{3}$

Step-by-step explanation:

Using the tangent ratio in the outer right triangle

and tan30° = $\frac{1}{\sqrt{3} }$ = $\frac{\sqrt{3} }{3}$

tan30° = $\frac{opposite}{adjacent}$ = $\frac{x}{15}$

Multiply both sides by 15

15 × tan30° = x

x = 15 × $\frac{\sqrt{3} }{3}$ = 5$\sqrt{3}$

Answer 2

The value of x is 7.5. It can be evaluated as follows:

Given that:

$a + b = 15$

y is perpendicular for both the composing right angled triangles.

The two of its angles are 60 degrees and 30 degrees respectively.

Calculations:

By using cotangent trigonometric ratios, we have:

$cot(60) = \dfrac{a}{y}\\\\\dfrac{1}{\sqrt{3}} = \dfrac{a}{y}\\\\cot(30) = \dfrac{b}{y}\\\\\sqrt{3} = \dfrac{b}{y}$

Adding both the values, we get:

$\sqrt{3} + \dfrac{1}{\sqrt{3}} = \dfrac{a+b}{y} = \dfrac{15}{y}\\\\\dfrac{4}{\sqrt{3}} = \dfrac{15}{y}\\\\y = \dfrac{15 \times \sqrt{3}}{4}$

And thus we have:  $a = \dfrac{y}{\sqrt{3}} = \dfrac{15}{4}$

From Pythagoras Theorem we have:

$x^2 = y^2 + a^2\\\\x^2 = \dfrac{225 \times 3}{16} + \dfrac{225}{16} = \dfrac{225}{4}\\x = \sqrt{\dfrac{225}{4}} = \dfrac{15}{2} = 7.5$

Thus, the value of x is 7.5

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