Right-angled triangles: แบบฝึกหัด

In triangle \(XYZ\), the angle \(\angle Y = 90^{\circ}\). The side lengths are \(XY = 7\) cm and \(YZ = 10\) cm.
Calculate the size of angle \(\angle YXZ\), giving your answer correct to 1 decimal place.

A vertical flagpole stands on level ground. From a point on the ground 18.5 m from the base of the flagpole, the angle of elevation to the top of the flagpole is \(38^\circ\).

Calculate the height of the flagpole, giving your answer correct to 3 significant figures.

A surveyor at a point A on horizontal ground measures the angle of elevation of the top of a vertical tower as \(25^{\circ}\). He then walks \(40\) m closer to the tower to point B, where the angle of elevation becomes \(40^{\circ}\).
Find the height of the tower, correct to 1 decimal place.

In a right-angled triangle \(ABC\), the side opposite to angle \(BAC\) is \(BC = 5 \text{ cm}\) and the hypotenuse is \(AC = 12 \text{ cm}\). Calculate the size of angle \(BAC\) correct to 1 decimal place.

A right-angled triangle has a hypotenuse of length \(13\text{ cm}\) and one leg of length \(5\text{ cm}\). Calculate the length of the third side.

A ship sails \(12\text{ km}\) North and then \(16\text{ km}\) East. Calculate the three-figure bearing of the ship from its starting point, to the nearest degree.

Two vertical poles of heights \(9\text{ m}\) and \(14\text{ m}\) stand on level ground. The distance between the bases of the two poles is \(12\text{ m}\). Calculate the straight-line distance between the tops of the two poles.

A vertical flagpole stands on horizontal ground. From a point on the ground \(25\text{ m}\) away from the base of the flagpole, the angle of elevation to the top of the flagpole is \(26^\circ\).
Calculate the height of the flagpole. Give your answer correct to 3 significant figures.

A person is standing at the top of a cliff of height \(45\text{ m}\) above sea level. From the top of the cliff, they observe a boat at sea.

(a) The angle of depression from the top of the cliff to the boat is \(19^\circ\). Calculate the direct distance from the observer to the boat. Give your answer correct to 3 significant figures.

(b) Calculate the horizontal distance from the base of the cliff to the boat. Give your answer correct to 3 significant figures.