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The diagram below shows two right-angled triangles - OCR - GCSE Maths - Question 12 - 2018 - Paper 1
Question 12
The diagram below shows two right-angled triangles. Prove that triangles PQS and QRS are similar. Not to scale.
Worked Solution & Example Answer:The diagram below shows two right-angled triangles - OCR - GCSE Maths - Question 12 - 2018 - Paper 1
Step 1
Show a pair of corresponding angles are equal
Answer
In triangle PQS, angle QPS is a right angle, and in triangle QRS, angle RQS is also a right angle. Since both triangles share angle QSR, we have:
- Angle QPS = Angle RQS = 90°
- Angle PQS = Angle QRS
Thus, the pairs of corresponding angles are equal.
Step 2
Show that the sides have the same ratio
Answer
To establish the similarity, we can calculate the ratios of the sides:
For triangle PQS:
- PS = 10 cm
- SQ = 4 cm
For triangle QRS:
- RS = 8 cm
- QS can be derived since triangle PQS and triangle QRS share the side SQ, which can be evaluated as the difference between PS and PQ:
- PQ = 4 cm, hence QS = PS - PQ = 10 cm - 4 cm = 6 cm
Thus, the ratios of the sides are:
- For triangle PQS: ( \frac{PQ}{PS} = \frac{4}{10} = 0.4 )
- For triangle QRS: ( \frac{QS}{RS} = \frac{6}{8} = 0.75 )
The ratio of corresponding sides must be equal for triangles to be similar.
Step 3