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Triangles
EXERCISE 6.2
CBSE_NCERT_TEXTBOOK
Grade 10
Question:
E and F are points on the sides PQ and PR respectively of a PQR. For each of the following cases, state whether EF || QR : (i) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm (ii) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm (iii) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
Step-by-Step Solution
Key Concept: Use the Basic Proportionality Theorem (also called Thales’ theorem): In Δ PQR, a line joining points E on PQ and F on PR is parallel to QR iff it divides the two sides proportionally, i.e., \(\frac{PE}{EQ}=\frac{PF}{FR}\) or equivalently \(\frac{PE}{PQ}=\frac{PF}{PR}\).
1. Recall the theorem – For Δ PQR, \(EF\parallel QR\) ⇔ \(\dfrac{PE}{EQ}=\dfrac{PF}{FR}\) (or \(\dfrac{PE}{PQ}=\dfrac{PF}{PR}\)).
2. Case (i)
- Compute \(\dfrac{PE}{EQ}=\dfrac{3.9}{3}=1.30\).
- Compute \(\dfrac{PF}{FR}=\dfrac{3.6}{2.4}=1.50\).
- Since \(1.30
eq 1.50\), the ratios are not equal.
- Conclusion: \(EF\) is not parallel to \(QR\).
3. Case (ii)
- Compute \(\dfrac{PE}{EQ}=\dfrac{4}{4.5}=\dfrac{8}{9}=0.888\ldots\).
- Compute \(\dfrac{PF}{FR}=\dfrac{8}{9}=0.888\ldots\).
- The two ratios are equal.
- Conclusion: \(EF\) is parallel to \(QR\).
4. Case (iii) – Here the whole side lengths are given, so use the alternative form of the theorem.
- \(\dfrac{PE}{PQ}=\dfrac{0.18}{1.28}=0.140625\).
- \(\dfrac{PF}{PR}=\dfrac{0.36}{2.56}=0.140625\).
- The ratios are equal.
- Conclusion: \(EF\) is parallel to \(QR\).
5. Summary of answers
- (i) No, \(EF\) is not parallel to \(QR\).
- (ii) Yes, \(EF\) is parallel to \(QR\).
- (iii) Yes, \(EF\) is parallel to \(QR\).
2. Case (i)
- Compute \(\dfrac{PE}{EQ}=\dfrac{3.9}{3}=1.30\).
- Compute \(\dfrac{PF}{FR}=\dfrac{3.6}{2.4}=1.50\).
- Since \(1.30
eq 1.50\), the ratios are not equal.
- Conclusion: \(EF\) is not parallel to \(QR\).
3. Case (ii)
- Compute \(\dfrac{PE}{EQ}=\dfrac{4}{4.5}=\dfrac{8}{9}=0.888\ldots\).
- Compute \(\dfrac{PF}{FR}=\dfrac{8}{9}=0.888\ldots\).
- The two ratios are equal.
- Conclusion: \(EF\) is parallel to \(QR\).
4. Case (iii) – Here the whole side lengths are given, so use the alternative form of the theorem.
- \(\dfrac{PE}{PQ}=\dfrac{0.18}{1.28}=0.140625\).
- \(\dfrac{PF}{PR}=\dfrac{0.36}{2.56}=0.140625\).
- The ratios are equal.
- Conclusion: \(EF\) is parallel to \(QR\).
5. Summary of answers
- (i) No, \(EF\) is not parallel to \(QR\).
- (ii) Yes, \(EF\) is parallel to \(QR\).
- (iii) Yes, \(EF\) is parallel to \(QR\).
Correct Answer: (i) No, (ii) Yes, (iii) Yes
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