Question

Find the missing side lengths. Write your answers in simplest radical form with the denominator rationalized.

Answer 1

Answer: x=5, y=(5
`√(3)`)/2

Explanation:

Using the 30-60-90 triangle rule, we know that the sides follow 1-(
`√(3)`/2)-2. The ratio between the side and the rule is 5/2, meaning that we know that y=(5
`√(3)`)/2 and x=5.

Answer 2

The missing side lengths of the triangle are:

Adjacent leg: -√6y / 2

Hypotenuse: -8√a² / 2

Finding the Missing Side Lengths:

To find the missing side lengths of the triangle, we can use the trigonometric ratios of right triangles. We know that the angle opposite the side length "y" is 90 degrees, which means that "y" is the hypotenuse of the triangle.

Finding the Adjacent Leg:

We know the opposite leg (5/2) and the hypotenuse (y). We can use the cosine function to find the adjacent leg:

cos 30° = adjacent leg / hypotenuse

cos 30° = adjacent leg / y

adjacent leg = cos 30° * y

adjacent leg = √3/2 * y

Finding the Hypotenuse:

We can also use the Pythagorean theorem to find the hypotenuse:

Pythagorean theorem: a² + b² = c²

a² = c² - b²

a² = y² - (√3/2 * y)²

a² = y² - (3y² / 4)

a² = y² / 4

y = √(a² * 4)

Simplifying the Radical Form:

We can simplify the radical form by multiplying the numerator and denominator by 2:

y = √((a² * 4) * 2 / 2)

y = √(2a² * 4) / √2

y = 2√a² * √4 / √2

y = 2√a² * 2 / √2

y = 4√a² / √2

Therefore, the missing side lengths of the triangle are:

Adjacent leg: √3/2 * y

Hypotenuse: 4√a² / √2

Rationalizing the Denominators:

We can rationalize the denominators of the missing side lengths by multiplying the numerator and denominator by the conjugate of the denominator.

Conjugate of √2: -√2

Conjugate of √3 / 2: -√3 / 2

Rationalizing the Denominators:

Adjacent leg: √3/2 * y * -√2 / -√2

Adjacent leg: -√6y / 2

Hypotenuse: 4√a² / √2 * -√2 / -√2

Hypotenuse: -8√a² / 2

Therefore, the missing side lengths of the triangle are:

Adjacent leg: -√6y / 2

Hypotenuse: -8√a² / 2