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P, Q, R and S are four points on a circle - Edexcel - GCSE Maths - Question 16 - 2022 - Paper 3
Question 16
P, Q, R and S are four points on a circle. PX and SQ are straight lines. Prove that triangle PQX and triangle SRV are similar.
Worked Solution & Example Answer:P, Q, R and S are four points on a circle - Edexcel - GCSE Maths - Question 16 - 2022 - Paper 3
Step 1
Prove that angle PQX equals angle SRV
Answer
To show that triangle PQX is similar to triangle SRV, we can start by examining the angles.
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Since P, Q, R, and S lie on the circumference of the circle, the angles subtended by the same arc are equal. Therefore, we have:
- Angle PQX = angle SRV (angles subtended by arc QX and arc RV respectively)
This shows one pair of corresponding angles are equal.
Step 2
Step 3
Identify the third pair of angles
Answer
Finally, we note that the angles at points Q and R:
- Angle PQX + Angle QXP = 180°
- Angle SRV + Angle RVS = 180°
This implies:
- Angle PQX + Angle RVS = 180° (since both sets of angles are on the same straight line)
Thus, angle QXP equals angle RVS, which suggests that both sets of angles sum to 180°, confirming the similarity as all angle pairs have been established as equal.
Therefore, triangle PQX is similar to triangle SRV by the AA (Angle-Angle) similarity criterion.