A right triangle kite has a right angle at vertex N, with side KN adjacent to angle K and side NM adjacent to angle M. If angle K is $30°$ and the hypotenuse KM is $10$ units, what are the lengths of sides KN and NM? KN = $?$ units, NM = $?$ units

• A. KN = $8.66$ units, NM = $5$ units
• B. KN = $5$ units, NM = $8.66$ units
• C. KN = $7$ units, NM = $7$ units
• D. KN = $10$ units, NM = $5$ units

Answer: (B)

Verified step by step guidance

1. Identify the given information: triangle KNM is a right triangle with a right angle at vertex N, angle K is 30°, and the hypotenuse KM is 10 units.
2. Recall that in a right triangle, the side opposite the right angle is the hypotenuse. Here, KM is the hypotenuse, so sides KN and NM are the legs adjacent to angles K and M respectively.
3. Use the definitions of sine and cosine for angle K (30°): \(\sin(30^\circ) = \frac{\text{opposite side}}{\text{hypotenuse}} = \frac{NM}{KM}\) \(\cos(30^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{KN}{KM}\)
4. Substitute the known values into the equations: \(\sin(30^\circ) = \frac{NM}{10}\) \(\cos(30^\circ) = \frac{KN}{10}\)
5. Solve each equation for the unknown sides: \(NM = 10 \times \sin(30^\circ)\) \(KN = 10 \times \cos(30^\circ)\) This will give the lengths of sides NM and KN.