Question

Find the missing side lengths. Write your answers in simplest radical form with the denominator rationalized. Please answer the best you can if you choose to answer, as I don't want to give away all my points for nothing.

Answer 1

Answer:

$x = 5$

$y = \frac{5}{2}\sqrt3$

Step-by-step explanation:

Required

Find x and y

From the triangle, we can see that x is the longest side (i.e. the hypotenuse)

The sin of an angle is:

$\sin(\theta) = \frac{opposite}{hypotenuse}$

The relationship between the given angle (30 degrees), x and 5/2 is:

$\sin(30) = \frac{5/2}{x}$

Cross Multiply:

$x * \sin(30) = \frac{5/2}{x} * x$

$x * \sin(30) = \frac{5}{2}$

Solve for x

$x = \frac{5}{2\sin(30)}$

$\sin(30) = 0.5$

So, the expression becomes

$x = \frac{5}{2*0.5}$

$x = \frac{5}{1}$

$x = 5$

To solve for y, we make use of Pythagoras theorem:

$x^2 = y^2 + \frac{5}{2}^2$

Substitute 5 for x

$5^2 = y^2 + \frac{5}{2}$

$25 = y^2 + \frac{25}{4}$

Solve for $y^2$

$y^2 = 25 - \frac{25}{4}$

$y^2 = \frac{100 - 25}{4}$

$y^2 = \frac{75}{4}$

Square root of both sides

$y = \sqrt{\frac{75}{4}}$

Express 75 as 25 * 3

$y = \sqrt{\frac{25 * 3}{4}}$

Split:

$y = \sqrt{\frac{25}{4}} * \sqrt3$

$y = \frac{5}{2} * \sqrt3$

$y = \frac{5}{2}\sqrt3$