Class 10 Mathematics

# Triangle

## Exercise 6.2 (NCERT) Part 1

Question 1: In the given figures, DE || BC. Find EC in first figure and AD in second figure.

**Solution:** In the first figure;

Δ ADE ~ Δ ABC (Because DE || BC)

Hence;

`(AD)/(DC)=(AE)/(EC)`

Or, `(1.5)/(3)=(1)/(EC)`

Or, `EC=(3)/(1.5)=2 cm`

Similarly, in the second figure;

Δ ADE ~ Δ ABC (Because DE || BC)

Hence;

`(AD)/(DC)=(AE)/(EC)`

Or, `(AD)/(7.2)=(1.8)/(5.4)`

Or, `AD=(1.8xx7.2)/(5.4)=2.4 cm`

Question 2: E and F are points on the sides PQ and PR respectively of a Δ PQR. For each of the following cases, state whether EF || QR.

(a) PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm

**Solution:** For EF || QR, the figure should fulfill following criterion;

`(PE)/(EQ)=(PF)/(FR)`

In this case;

`(PE)/(EQ)=(3.9)/(3)=1.3`

`(PF)/(FR)=(3.6)/(2.4)=3/2`

It is clear that;

`(PE)/(EQ)≠(PF)/(FR)`

Hence; EF and QR are not parallel.

(b) PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm

**Solution:** In this case;

`(PE)/(EQ)=(4)/(4.5)=8/9`

`(PF)/(FR)=8/9`

It is clear that;

`(PE)/(EQ)=(PF)/(FR)`

Hence; EF || QR

(c) PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm

**Solution:** In this case;

`(PE)/(EQ)=(0.18)/(1.28-0.18)=(0.18)/(1.10)=(9)/(55)`

`(PF)/(FR)=(0.36)/(2.56-0.36)=(0.36)/(2.20)=(9)/(55)`

It is clear that;

`(PE)/(EQ)=(PF)/(FR)`

Hence; EF || QR

Question 3: In the given figure, if LM || CB and LN || CD, prove that `(AM)/(AB)=(AN)/(AD)`

**Solution:** In Δ ABC and Δ AML;

Δ ABC ∼ Δ AML (because ML || BC)

Hence;

`(AM)/(AB)=(AL)/(AC)`Hence;

`(AN)/(AD)=(AL)/(AC)`From above two equations;

`(AM)/(AB)=(AN)/(AD)`Question 4: In the given figure, DE || AC and DF || AE. Prove that `(BF)/(FE)=(BE)/(EC)`

**Solution:** In Δ ABC and ΔDBE;

`(BE)/(EC)=(BD)/(BA)`

Because Δ ABC ∼ Δ DBE

Similarly, in Δ ABE and Δ DBF;

`(BF)/(FE)=(BD)/(BA)`

From above two equations, it is clear;

`(BF)/(FE)=(BE)/(EC)`

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Theorem (Part 1)

Theorem (Part 2)

Exercise 6.1

Exercise 6.2 (Part 2)

Exercise 6.3 (Part 1)

Exercise 6.3 (Part 2)

Exercise 6.3 (Part 3)

Exercise 6.4

Exercise 6.5 (Part 1)

Exercise 6.5 (Part 2)

Exercise 6.6 (Part 1)

Exercise 6.6 (Part 2)