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In an equilateral triangle ABC, P and Q are midpoints of sides AB and AC respectively, such that PQ || BC. If PQ = 5 cm, then find the length of BC.
- a)5 cm
- b)10 cm
- c)15 cm
- d)12 cm
Correct answer is option 'B'. Can you explain this answer?
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In an equilateral triangle ABC, P and Q are midpoints of sides AB and ...
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In an equilateral triangle ABC, P and Q are midpoints of sides AB and ...
Given information:
- Triangle ABC is equilateral.
- P and Q are midpoints of sides AB and AC respectively.
- PQ || BC.
- PQ = 5 cm.
To find:
The length of BC.
Solution:
We can solve this problem using the properties of midpoints and parallel lines.
1. Identifying Midpoints:
- P and Q are midpoints of sides AB and AC respectively.
- In an equilateral triangle, the midpoints of the sides are also the midpoints of the corresponding opposite sides.
- Therefore, BP = PC and CQ = QA.
2. Using Similarity:
- Since PQ || BC, we can use the similarity property of triangles.
- Triangle PBC is similar to triangle ABC (by AA similarity).
- Therefore, the corresponding sides are proportional.
3. Proportional Sides:
- BP:AB = PC:BC
- Since BP = PC (from midpoint property), we can rewrite the proportion as BP:AB = BP:BC
- Canceling BP on both sides, we get AB = BC.
- This means that BC is equal to the length of the side AB.
4. Finding BC:
- Given that PQ = 5 cm, we know that AB = 2PQ (as P is the midpoint of AB).
- Substituting the value of PQ, we have AB = 2(5) = 10 cm.
- Therefore, BC = AB = 10 cm.
Hence, the length of BC is 10 cm.
Therefore, option B is the correct answer.
- Triangle ABC is equilateral.
- P and Q are midpoints of sides AB and AC respectively.
- PQ || BC.
- PQ = 5 cm.
To find:
The length of BC.
Solution:
We can solve this problem using the properties of midpoints and parallel lines.
1. Identifying Midpoints:
- P and Q are midpoints of sides AB and AC respectively.
- In an equilateral triangle, the midpoints of the sides are also the midpoints of the corresponding opposite sides.
- Therefore, BP = PC and CQ = QA.
2. Using Similarity:
- Since PQ || BC, we can use the similarity property of triangles.
- Triangle PBC is similar to triangle ABC (by AA similarity).
- Therefore, the corresponding sides are proportional.
3. Proportional Sides:
- BP:AB = PC:BC
- Since BP = PC (from midpoint property), we can rewrite the proportion as BP:AB = BP:BC
- Canceling BP on both sides, we get AB = BC.
- This means that BC is equal to the length of the side AB.
4. Finding BC:
- Given that PQ = 5 cm, we know that AB = 2PQ (as P is the midpoint of AB).
- Substituting the value of PQ, we have AB = 2(5) = 10 cm.
- Therefore, BC = AB = 10 cm.
Hence, the length of BC is 10 cm.
Therefore, option B is the correct answer.
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