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ABC is a triangle - Edexcel - GCSE Maths - Question 23 - 2021 - Paper 3

Question 23

ABC is a triangle. D is the point on BC such that angle BAD = angle DAC = -x°. Prove that AB/BD = AC/DC.

Worked Solution & Example Answer:ABC is a triangle - Edexcel - GCSE Maths - Question 23 - 2021 - Paper 3

Step 1

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Answer

From triangle ABD, we can write the sine rule as:

$\frac{AB}{\sin \angle ADB} = \frac{BD}{\sin \angle ABD}$

Similarly, for triangle ACD:

$\frac{AC}{\sin \angle ADC} = \frac{DC}{\sin \angle DAC}$

Step 2

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Answer

The area of triangle ABD can be expressed as:

$\text{Area}_{ABD} = \frac{1}{2} AB \cdot BD \cdot \sin \angle ADB$

And for triangle ACD:

$\text{Area}_{ACD} = \frac{1}{2} AC \cdot DC \cdot \sin \angle ADC$

Step 3

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Answer

Since both triangles share the angle A (i.e., angle BAD + DAC = angle A), we can say:

$\frac{AB \cdot BD \cdot \sin \angle ADB}{AC \cdot DC \cdot \sin \angle ADC} = 1$

Step 4

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Answer

From the sine rule established earlier and the proportionality of areas:

$\frac{AB}{BD} = \frac{AC}{DC}$

Thus, we have proved the required relation.

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