The diagonal and the longer side of a rectangle make an angle of 43.2. If the longer side is 12.6 cm, find the length of the shorter side.

Here are the solutions to the problems from the image: 5. The diagonal and the longer side of a rectangle make an angle of 43.2. If the longer side is 12.6 cm, find the length of the shorter side. Step 1: Identify the sides and the angle. Let L be the longer side (adjacent to the angle) and W be the shorter side (opposite the angle). Given: L = 12.6 cm, = 43.2^. We use the tangent function: () = oppositeadjacent = (W)/(L) Step 2: Substitute the given values. (43.2^) = (W)/(12.6 cm) Step 3: Solve for W. W = 12.6 cm × (43.2^) W = 12.6 cm × 0.9391 W ≈ 11.833 cm The length of the shorter side is approximately 11.83 cm. 6. A rhombus has diagonals of length 12 cm and 7 cm respectively. Find the larger angle of the rhombus. Step 1: Understand the properties of a rhombus. The diagonals of a rhombus bisect each other at right angles. This forms four congruent right-angled triangles. The legs of these triangles are half the lengths of the diagonals. Half of diagonal 1: x = 12 cm2 = 6 cm. Half of diagonal 2: y = 7 cm2 = 3.5 cm. Step 2: Find half of the larger angle of the rhombus. Let be half of the larger angle of the rhombus. In one of the right-angled triangles, is the angle opposite the longer half-diagonal (x) and adjacent to the shorter half-diagonal (y). () = oppositeadjacent = (x)/(y) () = (6)/(3.5) () ≈ 1.7143 Step 3: Solve for and the larger angle. = (1.7143) ≈ 59.74^ The larger angle of the rhombus is 2. 2 ≈ 2 × 59.74^ 2 ≈ 119.48^ The larger angle of the rhombus is approximately 119.48^. 7. Find the angle of elevation to the top of a 62 m high building from point A, which is at ground level 155m from its base. What is the angle of depression from the top of the building to A? Step 1: Find the angle of elevation. Let h be the height of the building (62 m) and d be the distance from point A to the base (155 m). Let _E be the angle of elevation. (_E) = oppositeadjacent = (h)/(d) (_E) = 62 m155 m (_E) = 0.4 Step 2: Calculate _E. _E = (0.4) _E ≈ 21.80^ The angle of elevation is approximately 21.80^. Step 3: Find the angle of depression. The angle of depression from the top of the building to point A is equal to the angle of elevation from point A to the top of the building (alternate interior angles). Therefore, the angle of depression is also approximately 21.80^. 8. The angle of depression from the top of a 180 m high vertical cliff to a boat B is 26. Find how far the boat is from the base of the cliff. Step 1: Identify the knowns and unknowns. Let h be the height of the cliff (180 m). Let d be the distance from the boat to the base of the cliff (horizontal distance). The angle of depression from the top of the cliff to the boat is 26^. This means the angle of elevation from the boat to the top of the cliff is also 26^. We use the tangent function: (angle) = oppositeadjacent Step 2: Set up the equation. (26^) = (h)/(d) (26^) = 180 md Step 3: Solve for d. d = 180 m(26^) d = 180 m0.4877 d ≈ 369.09 m The boat is approximately 369.09 m from the base of the cliff. Last free one today — make it count tomorrow, or type /upgrade for unlimited.