LemonmathSelf-study sheet

Grade 9 · Lesson 4.2

Relations between sides and angles in a right-angled triangle, with applications

Name: Date: Time spent: min

Goals

Write and use the relations between the sides and angles of a right-angled triangle.
Solve a right-angled triangle given two sides, or one side and one acute angle.
Find real heights and distances without measuring them directly.

Remember

Triangle $ABC$ with a right angle at $A$ : $b=a\sin B=a\cos C$ , $c=a\sin C=a\cos B$ .
$b=c\tan B=c\cot C$ , $c=b\tan C=b\cot B$ .
Solving a right-angled triangle: Pythagoras' theorem, $\angle B+\angle C=90^\circ$ and the relations; at least one side must be known.
Real-life problems: draw a picture, find the right-angled triangle, choose the ratio, round at the last step.

Exercises

Exercise 1

Triangle $ABC$ has a right angle at $A$ , $BC=8$ cm and $\angle C=30^\circ$ . Find $AB$ and $AC$ .

Exercise 2

Triangle $ABC$ has a right angle at $A$ , $AC=10$ cm and $\angle C=42^\circ$ . Find $AB$ and $BC$ (rounded to two decimal places).

Exercise 3

Solve triangle $ABC$ with a right angle at $A$ , given $AB=9$ cm and $BC=15$ cm (angles to the nearest degree).

Exercise 4

Solve triangle $ABC$ with a right angle at $A$ , given $AC=4$ cm and $\angle B=50^\circ$ (sides to two decimal places).

Exercise 5

Triangle $ABC$ has a right angle at $A$ , $AB=2\sqrt3$ cm and $\angle B=60^\circ$ . Find the exact values of $AC$ and $BC$ .

Exercise 6

A $6$ m ladder leans against a wall and makes an angle of $70^\circ$ with the ground. How far is its foot from the wall, and how high is its top (rounded to two decimal places)?

Exercise 7

From the top of a lighthouse $40$ m above the sea, the keeper sees a boat at an angle of depression of $12^\circ$ . How far is the boat from the foot of the lighthouse (to the nearest metre)?

Exercise 8

When the sun's rays make an angle of $52^\circ$ with the ground, a tree casts a shadow $9$ m long on flat ground. Find the height of the tree (rounded to one decimal place).

Exercise 9

Hà is flying a kite. The string is straight and $50$ m long, and makes an angle of $38^\circ$ with the horizontal; Hà holds the string $1.2$ m above the ground.

a) How high is the kite above the ground (rounded to one decimal place)?

b) What is the horizontal distance from Hà's hand to the point directly below the kite (rounded to one decimal place)?

c) Hà lets out more string until it is $60$ m long, keeping the same angle. How many metres higher does the kite go (rounded to one decimal place)?

Exercise 10 Challenge

Two friends stand on the same straight line through the foot of a building, on the same side, $20$ m apart. The nearer one sees the top of the building at an angle of elevation of $45^\circ$ , the farther one at $30^\circ$ . Find the height of the building (ignore eye height; give an exact form, then round to two decimal places).

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. Triangle

ABC

has a right angle at

A

BC=a

and

CA=b

. Then

b=a\sin B

.B.

b=a\cos B

.C.

b=a\tan B

.D.

b=a\sin C

Question 2. Triangle

ABC

has a right angle at

A

and

AB=c

. Then

AC

equals

c\tan B

.B.

c\cot B

.C.

c\sin B

.D.

c\cos B

Question 3. Triangle

ABC

has a right angle at

A

BC=10

and

\angle B=30^\circ

. Then

AC

equals

5

.B.

5\sqrt3

.C.

10\sqrt3

.D.

20

Question 4. An

8

m electricity pole casts a shadow

8\sqrt3

m long on the ground. The angle between the sun's rays and the ground is

30^\circ

.B.

60^\circ

.C.

45^\circ

.D. about

0.58^\circ

Part II. True or false

Question 1. Triangle

ABC

has a right angle at

A

AB=6

cm and

\angle C=30^\circ

. Decide whether each statement is true or false.

a) $BC=12$ cm.	T / F
b) $AC=6\sqrt3$ cm.	T / F
c) $\angle B=30^\circ$ .	T / F
d) $AC=BC\sin C$ .	T / F

Part III. Short answer

Question 1. A

4

m ladder leans against a wall and makes an angle of

60^\circ

with the ground. How many metres above the ground is its top? (Give your answer to 1 decimal place.)

Question 2. A building casts a shadow

25

m long on the ground when the sun's rays make an angle of

40^\circ

with the ground. How tall is the building in metres? (Give your answer to the nearest whole number.)

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $c=a\sin C$ , $b=a\cos C$ ; $\sin30^\circ=\dfrac12$ , $\cos30^\circ=\dfrac{\sqrt3}2$ .

Steps:

$AB$ is opposite angle $C$ .
$AC$ is adjacent to angle $C$ .

Exercise 2.

What you need: $c=b\tan C$ ; $b=a\cos C\Rightarrow a=\dfrac{b}{\cos C}$ .

Steps:

$AB=AC\tan C$ .
$BC=\dfrac{AC}{\cos C}$ .

Exercise 3.

What you need: Pythagoras' theorem; $\sin C=\dfrac{AB}{BC}$ .

Steps:

Find $AC$ .
Find angle $C$ , then angle $B=90^\circ-\angle C$ .

Exercise 4.

What you need: $c=b\cot B$ ; $b=a\sin B$ .

Steps:

$\angle C=90^\circ-50^\circ$ .
$AB=AC\cot B=\dfrac{AC}{\tan B}$ ; $BC=\dfrac{AC}{\sin B}$ .

Exercise 5.

What you need: $\tan60^\circ=\sqrt3$ , $\cos60^\circ=\dfrac12$ (see trigonometric ratios of acute angles).

Steps:

$AC=AB\tan B$ .
$BC=\dfrac{AB}{\cos B}$ .

Exercise 6.

What you need: the ladder is the hypotenuse; the distance to the wall is adjacent to the $70^\circ$ angle.

Steps:

Distance $=6\cos70^\circ$ .
Height $=6\sin70^\circ$ .

Exercise 7.

What you need: the angle of depression equals the angle between the line of sight and the sea (alternate angles).

Steps:

In the right-angled triangle, the $40$ m side is opposite the $12^\circ$ angle at the boat.
Distance $=\dfrac{40}{\tan12^\circ}$ .

Exercise 8.

What you need: the height is opposite the $52^\circ$ angle and the shadow is adjacent to it.

Steps:

Draw the right-angled triangle: tree, shadow, sun's ray.
Height $=9\tan52^\circ$ .

Exercise 9.

What you need: the string is the hypotenuse; the height is opposite the $38^\circ$ angle.

Steps:

Height above the hand: $50\sin38^\circ$ ; add $1.2$ .
Horizontal distance: $50\cos38^\circ$ .
The rise: $(60-50)\sin38^\circ$ .

Exercise 10.

What you need: the distance to the foot of the building is $h\cot\alpha$ ; rationalising the denominator.

Steps:

Let $h$ be the height. The two distances are $h\cot45^\circ=h$ and $h\cot30^\circ=h\sqrt3$ .
Form the equation $h\sqrt3-h=20$ and solve it.

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