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In the diagram, AED and ABC are straight lines and BE is parallel to CD - OCR - GCSE Maths - Question 13 - 2020 - Paper 6

Question 13

In the diagram, AED and ABC are straight lines and BE is parallel to CD. The ratio of length AB to length BC is 2 : 3. Triangle ABE has an area of 8 cm². Work out ... show full transcript

Worked Solution & Example Answer:In the diagram, AED and ABC are straight lines and BE is parallel to CD - OCR - GCSE Maths - Question 13 - 2020 - Paper 6

Step 1

Find the ratio of triangle areas

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Answer

Since triangles ABE and ACD are similar due to the parallel lines, the ratio of their areas is the square of the ratio of their corresponding sides. Given the ratio of AB to BC is 2:3, the ratio AB to AC (which includes both AB and BC) is:

$\text{Ratio of } AB : AC = 2 : (2 + 3) = 2 : 5$

Thus, the ratio of the areas of triangles ABE to ACD is:

$\left( \frac{2}{5} \right)^2 = \frac{4}{25}$

Step 2

Calculate the area of triangle ACD

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Answer

Let the area of triangle ACD be represented as ( A_{ACD} ). We know:

$\frac{\text{Area of ABE}}{\text{Area of ACD}} = \frac{4}{25}$

Substituting the known area of triangle ABE (8 cm²):

$\frac{8}{A_{ACD}} = \frac{4}{25}$

Cross-multiplying gives:
$4 A_{ACD} = 8 \times 25$
$4 A_{ACD} = 200$
$A_{ACD} = \frac{200}{4} = 50$

Thus, the area of triangle ACD is 50 cm².

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