If ∠L≅∠M and both equal 90 degrees, ∠LKF=35 degrees, and KF=12 mm, then what is the length of MF to the nearest millimeter? (1 point) Responses 11 mm 11 mm 7 mm 7 mm 10 mm 10 mm 20 mm

Answer 1

To solve for the length of \( MF \), we can apply the properties of right triangles based on the given information.

1. Since \( ∠L \) and \( ∠M \) both equal 90 degrees, triangle \( LKF \) is a right triangle at \( K \).
2. We know one angle \( ∠LKF = 35^\circ \) and the side \( KF = 12 \) mm opposite to it.

Using the definition of the sine function in a right triangle, we can find the length of \( MF \) (which is adjacent to the angle \( ∠LKF \)) using the cosine function.

The relationship will be: \[ \cos(∠LKF) = \frac{{\text{{adjacent}}}}{{\text{{hypotenuse}}}} = \frac{MF}{KF} \]

Substituting the known values: \[ \cos(35^\circ) = \frac{MF}{12} \]

Now we need to solve for \( MF \): \[ MF = 12 \cdot \cos(35^\circ) \]

Using a calculator or cosine table: \[ \cos(35^\circ) \approx 0.8192 \]

Now calculate \( MF \): \[ MF \approx 12 \cdot 0.8192 \approx 9.8304 \]

Rounding this value to the nearest millimeter gives: \[ MF \approx 10 \text{ mm} \]

Therefore, the length of \( MF \) to the nearest millimeter is: \[ \boxed{10 \text{ mm}} \]