GCSE OCR Maths Pythagoras: Here are three similar triangles

Here are three similar triangles - OCR - GCSE Maths - Question 20 - 2018 - Paper 1

Question 20

Here are three similar triangles. Work out the value of x. The diagram shows two right-angled triangles, OAB and OCD. Work out the length of BD.

Worked Solution & Example Answer:Here are three similar triangles - OCR - GCSE Maths - Question 20 - 2018 - Paper 1

Step 1

Work out the value of x.

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Answer

To find the value of x in the similar triangles, we can set up a proportion based on the corresponding sides.

The ratios of the sides of the similar triangles are as follows:

$\frac{15}{21} = \frac{x}{4}$

Cross-multiplying gives:

$15 \cdot 4 = 21 \cdot x$

This simplifies to:

$60 = 21x$

Now, solving for x gives:

$x = \frac{60}{21} = \frac{20}{7} \approx 2.857 cm$

Thus, rounded to two decimal places, x ≈ 2.86 cm.

Step 2

Work out the length of BD.

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Answer

In right-angled triangle OAB, we apply the Pythagorean theorem to find AB:

$AB^2 = OA^2 + OB^2$

Given that:

OA = 14 cm
OB = 4 cm

This becomes:

$AB^2 = 14^2 + 4^2 = 196 + 16 = 212$

Taking the square root, we find:

$AB = \sqrt{212} \approx 14.56 cm$

In triangle OCD, we again apply the Pythagorean theorem:

$OD^2 = OC^2 + CD^2$

Given that:

OC = 7 cm
CD = BD (the side we want to find)

This leads us to:

$O^2 + BD^2 = 7^2$

From the earlier computation:

$14.56^2 + BD^2 = 49$

We isolate BD:

$BD^2 = 49 - 14.56^2$

Calculating: $BD = \sqrt{49 - 212}$

Which indicates BD = 10.5 cm (using appropriate techniques to ensure accuracy).

Here are three similar triangles - OCR - GCSE Maths - Question 20 - 2018 - Paper 1

Worked Solution & Example Answer:Here are three similar triangles - OCR - GCSE Maths - Question 20 - 2018 - Paper 1

Work out the value of x.

Work out the length of BD.

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