$ABC$ is a triangle in which $A\hat{B}C = 90^\circ$, $B\hat{A}C = 40^\circ$ and $BC = 10\,\text{cm}$. $P$ is the point on $AB$ such that $P\hat{C}B = 20^\circ$. Calculate (a) $PB$, \_\_\_\_\_\_ $\text{cm}$ (b) $AP$, \_\_\_\_\_\_ $\text{cm}$ (c) the perimeter of triangle $PBC$. \_\_\_\_\_\_ $\text{cm}$

(a) 3.64 · (b) 8.28 · (c) 24.3

Mathematics (Syllabus D) 4024/21 — October/November 2011

Cambridge O-Level · Calculator · worked solutions for every part, with the mark scheme

questions

100

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150

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Topics Algebra and Graphs · Geometry · Mensuration · Trigonometry · Probability · Number · +4 more

Q1TrigonometryFree sample

$ABC$ is a triangle in which $A\hat{B}C = 90^\circ$ , $B\hat{A}C = 40^\circ$ and $BC = 10\,\text{cm}$ .
$P$ is the point on $AB$ such that $P\hat{C}B = 20^\circ$ .

Fig. 1

Calculate

(a)

$PB$ ,

______ $\text{cm}$

DifficultyEasy

Worked solution

Approach

In right-angled triangle $PBC$ , the right angle is at $B$ , the angle at $C$ is $20^\circ$ , and the adjacent side $BC = 10\,\text{cm}$ . Use the tangent ratio to find the opposite side $PB$ .

Working

\tan 20^\circ = \frac{PB}{BC} = \frac{PB}{10}

PB = 10 \times \tan 20^\circ

PB = 10 \times 0.36397\ldots = 3.6397\ldots

Answer

PB = 3.64\,\text{cm}

Final answer

3.64

Detailed explanation

Walkthrough

We are given right-angled triangle $PBC$ with the right angle at $B$ . The angle $\angle PCB = 20^\circ$ and the adjacent side $BC = 10\,\text{cm}$ . We need to find the opposite side $PB$ . The tangent ratio relates the opposite and adjacent sides of a right-angled triangle: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ . Substituting the known values gives $\tan 20^\circ = \frac{PB}{10}$ , so $PB = 10 \tan 20^\circ$ . Evaluating this on a calculator gives approximately $3.6397$ , which rounds to $3.64\,\text{cm}$ to 3 significant figures.

Key Takeaways

The tangent ratio $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$ is used when you know one acute angle and the adjacent side, and need to find the opposite side in a right-angled triangle.

Common Mistakes

Using the wrong trigonometric ratio (e.g., sine or cosine instead of tangent).
Forgetting to round to the required accuracy (3 significant figures is standard unless stated otherwise).
Using degrees instead of radian mode on the calculator.

Things to Be Careful About

Ensure your calculator is in degree mode. The mark scheme accepts $10\tan 20^\circ$ as the method mark, so showing the substituted expression before the final value is important. Answers should be given to at least 3 significant figures.

Techniques used

identify the right-angled triangle PBCuse the tangent ratio to find PB

(b)

$AP$ ,

______ $\text{cm}$

DifficultyMedium-Easy

Worked solution

Approach

In right-angled triangle $ABC$ , the right angle is at $B$ , the angle at $A$ is $40^\circ$ , and the opposite side $BC = 10\,\text{cm}$ . Use the tangent ratio to find the total length $AB$ , then subtract $PB$ to find $AP$ .

Working

\tan 40^\circ = \frac{BC}{AB} = \frac{10}{AB}

AB = \frac{10}{\tan 40^\circ} = 10 \times \tan 50^\circ

AB = 10 \times 1.19175\ldots = 11.9175\ldots

AP = AB - PB = 11.9175\ldots - 3.6397\ldots = 8.2778\ldots

Answer

AP = 8.28\,\text{cm}

Final answer

8.28

Detailed explanation

Walkthrough

First, consider the larger right-angled triangle $ABC$ . The angle $\angle BAC = 40^\circ$ , the opposite side $BC = 10\,\text{cm}$ , and we need the adjacent side $AB$ . Using the tangent ratio: $\tan 40^\circ = \frac{10}{AB}$ , which gives $AB = \frac{10}{\tan 40^\circ}$ . This is equivalent to $10 \tan 50^\circ$ (since $\tan(90^\circ - \theta) = \cot \theta = \frac{1}{\tan \theta}$ ). Calculating this gives $AB \approx 11.9175\,\text{cm}$ . Since $P$ lies on $AB$ , we have $AP = AB - PB$ . Substituting the values: $AP = 11.9175 - 3.6397 = 8.2778\,\text{cm}$ , which rounds to $8.28\,\text{cm}$ .

Key Takeaways

When a point lies on a side of a triangle, you can find the segment lengths by calculating the total side length using trigonometry and then subtracting the known portion.

Common Mistakes

Using $\tan 40^\circ = \frac{AB}{10}$ instead of $\frac{10}{AB}$ (mixing up opposite and adjacent).
Carrying forward an unrounded value from part (a) incorrectly, or using a rounded value that introduces error.
Forgetting that $AP = AB - PB$ and instead trying to find $AP$ directly from triangle $APC$ without enough information.

Things to Be Careful About

Show the substituted expression $\frac{10}{\tan 40^\circ}$ or $10(\tan 50^\circ - \tan 20^\circ)$ to earn the method mark. Use unrounded values from previous parts in intermediate calculations to avoid rounding errors. Round the final answer to 3 significant figures.

Techniques used

use the tangent ratio to find the total length ABsubtract PB from AB to find AP

(c)

the perimeter of triangle $PBC$ .

______ $\text{cm}$

DifficultyMedium-Easy

Worked solution

Approach

In right-angled triangle $PBC$ , use the cosine ratio to find the hypotenuse $PC$ , then add $PB$ , $BC$ , and $PC$ to find the perimeter of triangle $PBC$ .

Working

\cos 20^\circ = \frac{BC}{PC} = \frac{10}{PC}

PC = \frac{10}{\cos 20^\circ}

PC = 10.6417\ldots

\text{Perimeter} = PB + BC + PC = 3.6397\ldots + 10 + 10.6417\ldots = 24.2814\ldots

Answer

\text{Perimeter} = 24.3\,\text{cm}

Final answer

24.3

Detailed explanation

Walkthrough

The perimeter of triangle $PBC$ is the sum of its three sides: $PB + BC + PC$ . We already know $PB \approx 3.6397\,\text{cm}$ and $BC = 10\,\text{cm}$ . We need to find $PC$ , the hypotenuse of right-angled triangle $PBC$ . Using the cosine ratio: $\cos 20^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{10}{PC}$ , so $PC = \frac{10}{\cos 20^\circ} \approx 10.6417\,\text{cm}$ . Adding the three sides: $3.6397 + 10 + 10.6417 = 24.2814\,\text{cm}$ , which rounds to $24.3\,\text{cm}$ .

Key Takeaways

The perimeter of a triangle is the sum of its three side lengths. When one side is unknown, use the appropriate trigonometric ratio (sine, cosine, or tangent) to find it before summing.

Common Mistakes

Using sine instead of cosine to find $PC$ (e.g., $\sin 20^\circ = \frac{10}{PC}$ is incorrect because $BC$ is adjacent to $20^\circ$ , not opposite).
Adding $AP$ instead of $PB$ when calculating the perimeter of triangle $PBC$ .
Rounding intermediate values too early, which can lead to answers like $24.2$ instead of $24.3$ .

Things to Be Careful About

The mark scheme awards method marks for finding $PC = \frac{10}{\cos 20^\circ}$ and for summing their answer to part (a) plus $10$ plus their $PC$ . Use unrounded values for $PB$ and $PC$ in the final addition to ensure accuracy. Round the final perimeter to 3 significant figures.

Techniques used

use the cosine ratio to find the hypotenuse PCadd the three side lengths to find the perimeter

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