Question

What is the measure in degrees of angle LMN in the figure shown?

Answer 1

The measure of
`$\angle MLN$` is approximately
`105^\circ$.`

The triangle in question is a 30-60-90 triangle. This means that the ratio of the sides is
`$1:√(3):2$`, and the smallest angle is
`30^\circ.`

To find the measure of
`\angle MLN$`, we can use the following approach:

Identify the sides: We are given that
`ML = 10√(3)$` and
`LN = 10√(6)$.`

Determine the ratio of the sides: Based on the 30-60-90 triangle properties, we know that
`MN = 2 \cdot ML = 20√(3)$.`

Use the law of cosines: The law of cosines states that for any triangle ABC, the following equation holds:

`$AC^2` =
`AB^2 + BC^2`
`- 2AB \cdot BC \cdot \cos(C)$.`

In this case, we can use it to find
`\cos(MLN)$:`

`$$\cos(MLN)`
`= (ML^2 + LN^2 - MN^2)/(2 \cdot ML \cdot LN)$$`

cos(MLN)
`= ((10√(3))^2 + (10√(6))^2 - (20√(3))^2)/(2 \cdot (10√(3)) \cdot (10√(6)))$$`

`$$\cos(MLN)`=
`(300 + 600 - 1200)/(200 \cdot 60) =`
`-(1)/(4)$$`

Find the angle: Since we know the cosine of
`$\angle MLN$`, we can find the angle itself using the inverse cosine function (arccosine):
`$$\angle MLN = \cos^(-1)(-(1)/(4))$$`

Using the calculator, we get
`\angle MLN \approx 105^\circ$.`