Right Triangles and Trigonometry: math practice, Grade 10 – download the PDF

1 Zip line cable ★★★

A zip line cable runs from the top of a \(12\) m pole straight to a point on level ground \(16\) m from the base of the pole. How long is the cable?

2 Missing leg ★★★

A right triangle has a hypotenuse of \(26\) cm and one leg of \(10\) cm. Find the other leg.

3 Is it a right triangle? ★★★

Decide whether each triangle is right, acute, or obtuse.

1. \(8,\ 15,\ 17\)
2. \(6,\ 7,\ 10\)

4 Isosceles right triangle ★★★

A 45°-45°-90° triangle is given.

1. Its legs measure \(6\) in. Find the hypotenuse.
2. Another such triangle has hypotenuse \(10\) in. Find its legs.

5 Half an equilateral triangle ★★★

In a 30°-60°-90° triangle, the short leg measures \(5\) cm. Find the long leg and the hypotenuse.

6 Naming the ratios ★★★

A right triangle has legs \(5\) and \(12\) and hypotenuse \(13\). Let \(A\) be the angle opposite the leg \(5\), and \(B\) the other acute angle. Write \(\sin\), \(\cos\), and \(\tan\) of \(A\) and of \(B\) as fractions.

7 True or false? ★★★

Say whether each statement is true or false, and justify.

1. \(\sin 35^\circ = \cos 55^\circ\).
2. For an acute angle \(A\), it is possible that \(\sin A = 1.2\).
3. For an acute angle \(A\), it is possible that \(\tan A = 3\).
4. \(\cos A = \dfrac{\text{adjacent}}{\text{opposite}}\).

8 Geometric means ★★★

Find the geometric mean of each pair of numbers. Give an exact value, then a decimal rounded to the nearest hundredth when needed.

1. \(4\) and \(25\)
2. \(6\) and \(8\)

9 Altitude to the hypotenuse ★★★

In right triangle \(ABC\) (right angle at \(C\)), the altitude from \(C\) meets the hypotenuse at \(D\), with \(AD = 3\) and \(DB = 12\). Find \(CD\), \(AB\), \(AC\), and \(BC\) (exact values, then decimals to the nearest hundredth).

10 Water slide ★★★

A water slide is \(18\) m long and makes a \(35^\circ\) angle with the horizontal. Find its height and its horizontal length, to the nearest hundredth of a meter.

11 Angles of a right triangle ★★★

A right triangle has legs \(8\) in and \(15\) in. Find both acute angles to the nearest tenth of a degree.

12 Wheelchair ramp ★★★

A ramp rises \(2.5\) ft over a horizontal run of \(30\) ft (figure not to scale). Find the angle the ramp makes with the ground, and the length of the ramp.

13 Height of a tower ★★★

Standing \(40\) m from the foot of a tower, you see its top at an angle of elevation of \(52^\circ\). Your eyes are \(1.6\) m above the ground. How tall is the tower, to the nearest tenth of a meter?

14 Lighthouse and boat ★★★

From the top of a lighthouse \(45\) m above sea level, the angle of depression to a boat is \(18^\circ\). How far is the boat from the base of the lighthouse (nearest meter)?

15 Law of Sines ★★★

In triangle \(ABC\), \(A = 48^\circ\), \(B = 65^\circ\), and \(a = 12\). Find \(C\), \(b\), and \(c\) to the nearest hundredth.

16 Law of Cosines ★★★

Two hiking trails leave the same trailhead and form a \(38^\circ\) angle. One is \(9\) mi long and the other is \(14\) mi long. How far apart are their far ends?

17 Which law? ★★★

For each situation, name the best first tool (Law of Sines or Law of Cosines) and say why. You do not need to solve it.

1. \(a = 9\), \(b = 12\), \(c = 15\); find the largest angle.
2. \(A = 35^\circ\), \(B = 80^\circ\), \(b = 14\); find \(a\).
3. \(a = 7\), \(b = 10\), \(C = 48^\circ\); find \(c\).
4. \(a = 20\), \(A = 30^\circ\), \(b = 28\); find \(B\).

18 Planes leaving an airport ★★★

Two planes leave an airport at the same time on paths that form a \(70^\circ\) angle. One flies at \(300\) mph and the other at \(240\) mph. How far apart are they after \(2\) hours, to the nearest mile?

19 Two angles of elevation ★★★

From point \(P\), the angle of elevation to the top of a building is \(28^\circ\). After walking \(60\) ft straight toward the building to point \(Q\), the angle of elevation is \(41^\circ\). Find the height \(h\) of the building to the nearest tenth of a foot.

20 Special triangles with algebra ★★★

1. An equilateral triangle has side \(12\) cm. Find its height and its area (exact values, then decimals to the nearest hundredth).
2. A square has a diagonal of \(20\) in. Find its side and its area.
3. A 30°-60°-90° triangle has a long leg of \(9\) ft. Find the short leg and the hypotenuse.

21 The ambiguous case ★★★

In triangle \(ABC\), \(a = 10\), \(b = 14\), and \(A = 30^\circ\). Show that two triangles are possible and find the remaining parts of each, to the nearest tenth.

22 Proving an identity ★★★

1. In a right triangle with legs \(a\), \(b\) and hypotenuse \(c\), let \(\theta\) be the angle opposite \(a\). Prove that \(\sin^2\theta + \cos^2\theta = 1\).
2. If \(\sin\theta = \dfrac{3}{5}\) and \(\theta\) is acute, find \(\cos\theta\) and \(\tan\theta\).
3. If \(\tan\theta = \dfrac{5}{12}\), find \(\sin\theta\) and \(\cos\theta\).

23 A triangular lot ★★★

A triangular lot has sides of \(120\) ft, \(150\) ft, and \(200\) ft. Is the lot a right triangle? Find all three angles to the nearest hundredth of a degree and check that their sum is \(180^\circ\).