Trigonometry in Quadrilaterals and Pyramids Practice

A right-angled trapezium has parallel bases of lengths 10 cm and 6 cm, and a perpendicular height of 3 cm. Find the length of the non-perpendicular slanted side.

5 cm. Drop a height to form a right triangle with legs of 3 cm and 4 cm (10 - 6). By the Pythagorean theorem: square root of (3^2 + 4^2) = square root of 25 = 5 cm.

Using the trapezium from problem 1, calculate the measure of the acute interior angle between the slanted side and the longer base.

Approximately 36.87 degrees. In the right triangle, tan(theta) = 3 / 4 = 0.75. Taking arctan(0.75) gives 36.87 degrees.

An isosceles trapezium has bases of 14 cm and 6 cm, and slanted legs of 5 cm each. Find the height of the trapezium.

3 cm. Dropping heights from both top vertices splits the excess length evenly: (14 - 6) / 2 = 4 cm for each side triangle. In the right triangle with hypotenuse 5 cm and base 4 cm, height = square root of (5^2 - 4^2) = square root of 9 = 3 cm.

A parallelogram has sides of length 7 cm and 9 cm, with an interior acute angle of 60 degrees. Find the length of the shorter diagonal.

Approximately 8.19 cm. Use the cosine rule in the triangle with sides 7 and 9 containing the 60-degree angle: d^2 = 7^2 + 9^2 - 2(7)(9)cos(60) = 49 + 81 - 126(0.5) = 130 - 63 = 67. The square root of 67 is about 8.19 cm.

A pyramid has a square base with sides of 8 cm and a vertical height of 7 cm. Calculate the length of the segment from the center of the base to any base vertex.

Approximately 5.66 cm (or 4 times the square root of 2). The base diagonal is 8 * sqrt(2) ≈ 11.31 cm. Half of this diagonal is 4 * sqrt(2) ≈ 5.66 cm.

Using the pyramid from problem 5, find the angle between a lateral edge and the base.

Approximately 51.04 degrees. In the right triangle formed by the vertical height, the half-diagonal, and the lateral edge: tan(theta) = 7 / (4 * sqrt(2)) ≈ 7 / 5.657 ≈ 1.2374. theta = arctan(1.2374) ≈ 51.04 degrees.

For the same pyramid, calculate the angle between a lateral triangular face and the base.

Approximately 60.26 degrees. The run from the center to the midpoint of an edge is half the side length: 8 / 2 = 4 cm. In the vertical slice triangle: tan(phi) = 7 / 4 = 1.75. phi = arctan(1.75) ≈ 60.26 degrees.