Question

1) Suppose a rhombus has 12 cm sides and a 30° angle. Find the distance between the pair of opposite sides.

2) In rectangle KLMN, the angle bisector of ∠NKM intersects the longer side at point P. The measure of ∠KML is equal to 54°. Find the measure of ∠KPM.

Answer 1

1) 6 cm

2) 117°

Explanation:

1) Draw a picture of the rhombus. The distance between opposite sides is the height of the rhombus. If we draw the height at the vertex, we get a right triangle. Using trigonometry:

sin 30° = h / 12

h = 12 sin 30°

h = 6 cm

2) Draw a picture of the rectangle.

∠KML is the angle the diagonal makes with the shorter side ML. This angle is 54°. ∠NKM is the angle the diagonal makes with the shorter side NK. ∠KML and ∠NKM are alternate interior angles, so m∠NKM = 54°.

The angle bisector of angle ∠NKM divides the angle into two equal parts and intersects the longer side NM at point P. So m∠PKM = 27°.

KLMN is a rectangle, so it has right angles. That means ∠KML and ∠KMN are complementary. So m∠KMN = 36°.

We now know the measures of two angles of triangle KPM. Since angles of a triangle add up to 180°, we can find the measure of the third angle:

m∠KPM + 36° + 27° = 180°

m∠KPM = 117°