and are points on the sides and respectively of a . For each of the following cases, state whether : (i) , , , and (ii) , , , and (iii) , , , and

Understand the Question According to the Converse of Basic Proportionality Theorem (BPT), if a line divides two sides of a triangle in the same ratio, then the line must be parallel to the third side. For with points on and on , if: We evaluate and compare these ratios for each given case. Part (i) , , , and Step 1 · Calculate and Compare the Ratios Since (), by the converse of BPT, is not parallel to . Answer > (i) is not parallel to Part (ii) , , , and Step 1 · Calculate and Compare the Ratios Since , by the converse of BPT, . Answer > (ii) Part (iii) , , , and Step 1 · Calculate and Compare the Ratios Since , by the converse of BPT, . Answer > (iii) Common Mistakes Inverting Segment Ratios: Comparing with instead of . The order of segments from vertex downward must be consistent on both sides. Segment vs. Full Side Confusion: In part (iii), and are entire side lengths, not segment lengths and . Both and are valid formulations of BPT.

Triangles | Exercise 6.2

Question 2

$E$ and $F$ are points on the sides $PQ$ and $PR$ respectively of a $\Delta PQR$ . For each of the following cases, state whether $EF \parallel QR$ :

(i) $PE = 3.9\text{ cm}$ , $EQ = 3\text{ cm}$ , $PF = 3.6\text{ cm}$ , and $FR = 2.4\text{ cm}$

(ii) $PE = 4\text{ cm}$ , $QE = 4.5\text{ cm}$ , $PF = 8\text{ cm}$ , and $RF = 9\text{ cm}$

(iii) $PQ = 1.28\text{ cm}$ , $PR = 2.56\text{ cm}$ , $PE = 0.18\text{ cm}$ , and $PF = 0.36\text{ cm}$

Solution

Understand the Question

According to the Converse of Basic Proportionality Theorem (BPT), if a line divides two sides of a triangle in the same ratio, then the line must be parallel to the third side.
For $\Delta PQR$ with points $E$ on $PQ$ and $F$ on $PR$ , $EF \parallel QR$ if: $\dfrac{PE}{EQ} = \dfrac{PF}{FR} \quad \text{or} \quad \dfrac{PE}{PQ} = \dfrac{PF}{PR}$
We evaluate and compare these ratios for each given case.

(i) $PE = 3.9\text{ cm}$ , $EQ = 3\text{ cm}$ , $PF = 3.6\text{ cm}$ , and $FR = 2.4\text{ cm}$

Step 1 · Calculate and Compare the Ratios

$\dfrac{PE}{EQ} = \dfrac{3.9}{3} = 1.3$

$\dfrac{PF}{FR} = \dfrac{3.6}{2.4} = 1.5$

Since $\dfrac{PE}{EQ} \neq \dfrac{PF}{FR}$ ( $1.3 \neq 1.5$ ), by the converse of BPT, $EF$ is not parallel to $QR$ .

Answer

(i) $EF$ is not parallel to $QR$

(ii) $PE = 4\text{ cm}$ , $QE = 4.5\text{ cm}$ , $PF = 8\text{ cm}$ , and $RF = 9\text{ cm}$

Step 1 · Calculate and Compare the Ratios

$\dfrac{PE}{QE} = \dfrac{4}{4.5} = \dfrac{8}{9}$

$\dfrac{PF}{RF} = \dfrac{8}{9}$

Since $\dfrac{PE}{QE} = \dfrac{PF}{RF} = \dfrac{8}{9}$ , by the converse of BPT, $EF \parallel QR$ .

Answer

(ii) $EF \parallel QR$

(iii) $PQ = 1.28\text{ cm}$ , $PR = 2.56\text{ cm}$ , $PE = 0.18\text{ cm}$ , and $PF = 0.36\text{ cm}$

Step 1 · Calculate and Compare the Ratios

$\dfrac{PE}{PQ} = \dfrac{0.18}{1.28} = \dfrac{18}{128} = \dfrac{9}{64}$

$\dfrac{PF}{PR} = \dfrac{0.36}{2.56} = \dfrac{36}{256} = \dfrac{9}{64}$

Since $\dfrac{PE}{PQ} = \dfrac{PF}{PR} = \dfrac{9}{64}$ , by the converse of BPT, $EF \parallel QR$ .

Answer

(iii) $EF \parallel QR$

Common Mistakes

Inverting Segment Ratios: Comparing $\dfrac{PE}{EQ}$ with $\dfrac{FR}{PF}$ instead of $\dfrac{PF}{FR}$ . The order of segments from vertex $P$ downward must be consistent on both sides.
Segment vs. Full Side Confusion: In part (iii), $PQ$ and $PR$ are entire side lengths, not segment lengths $EQ$ and $FR$ . Both $\dfrac{PE}{EQ} = \dfrac{PF}{FR}$ and $\dfrac{PE}{PQ} = \dfrac{PF}{PR}$ are valid formulations of BPT.

Question 2

More questions in Exercise 6.2