Question

PLEASE!!!!Solving for Measures of Right Triangles: Use the diagram to complete the statements. The measure of angle L is °. The trigonometric ratio that uses ∠M and LN to solve for NM is . The length of NM, to the nearest tenth, is approximately .

Answer 1

1. 70

2. tangent

3. 57.7

Explanation:

Answer 2

1) The measure of angle L is 70°
The measure of all the angles of a triangle add up to 180°. You're given two angle measurements: 90° (aka the right angle) and 20° (measure of ∠M). The third angle must be 180°-90°-20°= 70°.

2) The trigonometric ratio is
When you have a right triangle, you can use sine, cosine, and tangent to figure out the lengths of the sides or angle measurements if given enough information.

Remember SOHCAHTOA!
sinθ =
`(opposite)/(hypotenuse)`
cosθ =
`(adjacent)/(hypotenuse)`
tanθ =
`(opposite)/(adjacent)`
where θ is the angle you determine your adjacent and opposite sides from.

The adjacent side is next to the angle θ (and isn't the hypotenuse) and the opposite side is across from the angle θ. The hypotenuse is the longest side of the triangle and directly across from the 90° angle.

If you look at your picture, LN is opposite ∠M and NM (the angle you're looking for) is adjacent to ∠M, because it isn't the hypotenuse. That means the trigonometric ratio you need is tangent, since tangent uses both opposite and adjacent.

3) The length of NM is 57.7.
Finally solve for the length of NM using tanθ =
`(opposite)/(adjacent)`. You will be solving for the adjacent side, so start off by rearranging the tangent equation to solve for adjacent:
tanθ =
`(opposite)/(adjacent)`
adjacent(tanθ) = opposite
adjacent =
`(opposite)/(tan\theta)`

Now, just plug in the values for θ and the length of the opposite side LN:

`adjacent = (opposite)/(tan\theta) \\ adjacent = (21)/(tan20) \\ adjacent = (21)/(0.364) \\ adjacent = 57.7`