 # Finding Lengths Using the Cosine Rule

In this worksheet, students find lengths using the cosine rule. Key stage:  KS 4

Curriculum topic:  Geometry and Measures

Curriculum subtopic:  Know the Exact Values of Sin, Cos and Tan

Difficulty level:   ### QUESTION 1 of 10

This worksheet is about finding lengths using the  cosine rule.

Look at  triangle ABC below The cosine rule states that:

a2 = b2 + c2 - 2bc cosA

We use the cosine rule when we are dealing with two sides of a triangle and the angle between these two sides.

Example:

In  triangle PQR, which is not drawn to scale, PQ = 7 cm, QR = 22 cm, and ∠PQR = 145º.

Find the length PR to 3 sig figs. Using the cosine rule,

PR2 = 72 + 222 - 2 x 7 x 22 x cos145º

= 49 + 484 - 308 x (-0.819)

= 785.3

PR = √785.3 ≈ 28.0 cm

In  triangle PQR, which is not drawn to scale,

PQ = 8 cm, PR = 13 cm, and ∠QPR = 30º.

Use the cos rule to find the length QR to 3 sig figs.

(just write the number) In  triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and ∠PQR = 132º.

Use the cos rule to find the length PR to 3 sig figs.

(just write the number) In  triangle PQR, which is not drawn to scale,

QR = 10 cm, PR = 14 cm, and ∠PRQ = 32º.

Use the cos rule to find the length PQ to 3 sig figs.

(just write the number) In  triangle PQR, which is not drawn to scale,

PQ = 12 cm, PR = 22 cm, and ∠QPR = 35º.

Use the cos rule to find the length QR to 3 sig figs.

(just write the number) In  triangle ABC, which is not drawn to scale, AB = 12 cm, BC = 7cm, and ∠ABC = 75º.

Use the cos rule to find the length AC to 3 sig figs.

(just write the number)

In  triangle ABC, which is not drawn to scale, AC = 9 cm, BC = 7cm, and ∠ACB = 68º.

Use the cos rule to find the length AB to 3 sig figs.

(just write the number)

In  triangle ABC, which is not drawn to scale, AC = 10 cm, AB = 8.5cm, and ∠BAC = 43º.

Use the cos rule to find the length BC to 3 sig figs.

(just write the number)

In  triangle ABC, which is not drawn to scale, AC = 9 cm, BC = 9cm, and ∠ACB = 60º.

Use the cos rule to find the length AB.

(just write the number)

In  triangle ABC, which is not drawn to scale, a = 8 cm, b = 9.5cm, and ∠C = 63º.

Use the cos rule to find the length c to 3 sig figs.

(just write the number)

In  triangle ABC, which is not drawn to scale, c = 9.35 cm, b = 9.5cm, and ∠A = 59º.

Use the cos rule to find the length a to 3 sig figs.

(just write the number)

• Question 1

In  triangle PQR, which is not drawn to scale,

PQ = 8 cm, PR = 13 cm, and ∠QPR = 30º.

Use the cos rule to find the length QR to 3 sig figs.

(just write the number) 7.27
EDDIE SAYS
QR² = 8² + 13² - 2 × 8 × 13 × cos 30°
• Question 2

In  triangle PQR, which is not drawn to scale,

PQ = 9 cm, QR = 10 cm, and ∠PQR = 132º.

Use the cos rule to find the length PR to 3 sig figs.

(just write the number) 17.4
EDDIE SAYS
PR² = 9² + 10² - 2 × 9 × 10 × cos 132°
• Question 3

In  triangle PQR, which is not drawn to scale,

QR = 10 cm, PR = 14 cm, and ∠PRQ = 32º.

Use the cos rule to find the length PQ to 3 sig figs.

(just write the number) 7.65
EDDIE SAYS
PQ² = 10² + 14² - 2 × 10 × 14 × cos 32°
• Question 4

In  triangle PQR, which is not drawn to scale,

PQ = 12 cm, PR = 22 cm, and ∠QPR = 35º.

Use the cos rule to find the length QR to 3 sig figs.

(just write the number) 14.0
EDDIE SAYS
QR² = 12² + 22² - 2 × 12 × 22 × cos 35°
• Question 5

In  triangle ABC, which is not drawn to scale, AB = 12 cm, BC = 7cm, and ∠ABC = 75º.

Use the cos rule to find the length AC to 3 sig figs.

(just write the number)

12.2
EDDIE SAYS
AC² = 12² + 7² - 2 × 12 × 7 × cos 75°
• Question 6

In  triangle ABC, which is not drawn to scale, AC = 9 cm, BC = 7cm, and ∠ACB = 68º.

Use the cos rule to find the length AB to 3 sig figs.

(just write the number)

9.10
EDDIE SAYS
AB² = 9² + 7² - 2 × 9 × 7 × cos 68°
• Question 7

In  triangle ABC, which is not drawn to scale, AC = 10 cm, AB = 8.5cm, and ∠BAC = 43º.

Use the cos rule to find the length BC to 3 sig figs.

(just write the number)

6.92
EDDIE SAYS
BC² = 10² + 8.5² - 2 × 10 × 8.5 × cos 43°
• Question 8

In  triangle ABC, which is not drawn to scale, AC = 9 cm, BC = 9cm, and ∠ACB = 60º.

Use the cos rule to find the length AB.

(just write the number)

9
9.00
EDDIE SAYS
AB² = 9² + 9² - 2 × 9 × 9 × cos 60°
This is an equilateral triangle of side 9cm
• Question 9

In  triangle ABC, which is not drawn to scale, a = 8 cm, b = 9.5cm, and ∠C = 63º.

Use the cos rule to find the length c to 3 sig figs.

(just write the number)

9.23
EDDIE SAYS
c² = 8² + 9.5² - 2 × 8 × 9.5 × cos 63°
• Question 10

In  triangle ABC, which is not drawn to scale, c = 9.35 cm, b = 9.5cm, and ∠A = 59º.

Use the cos rule to find the length a to 3 sig figs.

(just write the number)