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ABC and ACD are right-angled triangles - Edexcel - GCSE Maths - Question 13 - 2022 - Paper 3

Question 13

ABC and ACD are right-angled triangles. DC = 8 cm Angle ADC = 45° Angle ABC = 20° Work out the length of AB. Give your answer correct to 3 significant figures.

Worked Solution & Example Answer:ABC and ACD are right-angled triangles - Edexcel - GCSE Maths - Question 13 - 2022 - Paper 3

Step 1

Work out the length of AC using trigonometric relationships

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Answer

In triangle ACD, we can use the definition of tangent because we know the angle and the opposite side. The formula is:

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Here,[ \tan(45°) = \frac{CD}{AC} ][ \tan(45°) = 1 \Rightarrow CD = AC \Rightarrow AC = 8 , cm ]

Step 2

Work out the length of AB using the sine rule

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Answer

In triangle ABC, we can now use the sine rule to find AB. We first need to find angle CAB. Since the angles in a triangle add up to 180°:

[ \text{Angle CAB} = 180° - 45° - 20° = 115° ]

Using the sine rule:

$\frac{AB}{\sin(20°)} = \frac{AC}{\sin(115°)}$

Substituting AC = 8 cm:

$AB = \frac{8 \cdot \sin(20°)}{\sin(115°)}$

Now calculating: [ AB \approx \frac{8 \cdot 0.3429}{0.9063} \approx 3.025 , cm ]

Rounding to 3 significant figures, we have: [ AB \approx 3.03 , cm ]

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