Question

Please help i dont know how to do this

Answer 1

CD: 8.5

m<D: 20.6°

m<C: 69.4°

Explanation:

CD:

The first thing it wants us to do is find the length of CD. Since the triangle shown is a right triangle, we can use Pythagorean theorem (
`a^(2)+b^(2)=c^(2)`) to solve for the missing length. It's important to remember that when using the Pythagorean theorem, c is the hypotenuse.

`a^(2)+b^(2)=c^(2)\\3^(2)+8^(2)=c^(2)\\9 + 64 = c^(2)\\73 = c^(2)\\√(73) =c`

Since our answer is no an integer, we must turn it into a decimal.

`√(73)` ≈ 8.544003745 ≈ 8.5

m<D:

Now, they want us to find the measure of <D. To do this, we will need to use trig functions (sine, cosine, tangent). To help us determine which trig function to use, we can remember the acronym SOH CAH TOA. This acronym tells us that sine is equal to opposite divided by hypotenuse, cosine is equal to adjacent divided by hypotenuse, and tangent is equal to opposite divided by adjacent. Since we do the hypotenuse and sides adjacent and opposite of <D, we can choose whichever trig function we want. For this problem, we will use tangent, so we can avoid using a rounded number, 8.5, as one of our sides.

Tan(D) = opposite / adjacent

Tan(D) = 3 / 8 [Take the tan inverse of both sides}

`Tan^(-1)(Tan(D))=Tan^(-1)(3/8)` [Simplify]

`D=Tan^(-1)(3/8)` [Solve]

D ≈ 20.55604522

D ≈ 20.6°

m<C:

Lastly, we must find the last unknown angle on the triangle. Since all angles on a triangle total 180°, if know that <C+<D+<E=180°. Let's solve this equation.

<C+<D+<E=180°

<C + 20.6 + 90 = 180 [Add]

<C + 110.6 = 180 [Subtract]

<C = 180 - 110.6 [Solve]

<C = 69.4°

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