Trigonometry in Isosceles Triangles and Quadrilaterals

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Designed for 10th-grade students. Draw an altitude or diagonals to create right triangles, then solve each problem. Round side lengths and angles to one decimal place.

An isosceles triangle has two equal sides of length 13 cm and a base of 10 cm. Find the length of the altitude drawn to the base.
An isosceles triangle has legs of length 15 cm and a base angle of 40 degrees. Calculate the length of the altitude.
An isosceles triangle has a base of 16 cm and an altitude of 6 cm. Find the measure of the base angles.
A rectangle measures 8 cm wide and 6 cm high. Find the acute angle between the diagonal and the 8 cm base.
For the same 8 cm by 6 cm rectangle, find the acute angle between the two diagonals.
The diagonals of a rhombus measure 12 cm and 16 cm. Find the length of each side of the rhombus.
A rhombus has side lengths of 10 cm and one diagonal of 12 cm. Find the measure of the larger interior angle of the rhombus.

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Trigonometry in Isosceles Triangles and Quadrilaterals

Answer key

An isosceles triangle has two equal sides of length 13 cm and a base of 10 cm. Find the length of the altitude drawn to the base.
12 cm (Half the base is 5 cm. Using the Pythagorean theorem: 13^2 - 5^2 = 169 - 25 = 144; sqrt(144) = 12 cm).
An isosceles triangle has legs of length 15 cm and a base angle of 40 degrees. Calculate the length of the altitude.
9.6 cm (sin(40) = h / 15, so h = 15 * sin(40) ≈ 9.64 cm).
An isosceles triangle has a base of 16 cm and an altitude of 6 cm. Find the measure of the base angles.
36.9 degrees (Half the base is 8 cm. tan(theta) = 6 / 8 = 0.75; arctan(0.75) ≈ 36.87 degrees).
A rectangle measures 8 cm wide and 6 cm high. Find the acute angle between the diagonal and the 8 cm base.
36.9 degrees (tan(theta) = 6 / 8 = 0.75; arctan(0.75) ≈ 36.87 degrees).
For the same 8 cm by 6 cm rectangle, find the acute angle between the two diagonals.
73.7 degrees (The diagonals form an isosceles triangle with base 6 cm and legs 5 cm, or double the complementary angle: 180 - 2 * 53.13 = 73.74 degrees).
The diagonals of a rhombus measure 12 cm and 16 cm. Find the length of each side of the rhombus.
10 cm (Diagonals bisect perpendicularly into legs of 6 cm and 8 cm. Side = sqrt(6^2 + 8^2) = sqrt(100) = 10 cm).
A rhombus has side lengths of 10 cm and one diagonal of 12 cm. Find the measure of the larger interior angle of the rhombus.
106.3 degrees (One right triangle leg is 6 cm, hypotenuse is 10 cm. cos(half-angle) = 6/10 = 0.6; half-angle = 53.13 degrees; full angle = 2 * 53.13 = 106.26 degrees).