E and F are points on the sides PQ and PR respectively of ΔPQR. For each of the following, state whether EF || QR or not?(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 = 1.8 cm and PF = 3.6 cm
Answers
SOLUTION :
(i)Given :
PE = 3.9 cm, EQ = 3 cm ,PF = 3.6 cm, FR = 2,4 cm
In ΔPQR, E and F are two points on side PQ and PR respectively.
∴ PE/EQ = 3.9/3 = 39/30 = 13/10 = 1.3
[By using Basic proportionality theorem]
And, PF/FR = 3.6/2.4 = 36/24 = 3/2 = 1.5
So, PE/EQ ≠ PF/FR
Hence, EF is not || QR.
[By Converse of basic proportionality theorem]
(ii) Given :
PE = 4 cm, QE = 4.5 cm, PF = 8cm, RF = 9cm
∴ PE/QE = 4/4.5 = 40/45 = 8/9
[By using Basic proportionality theorem]
And, PF/RF = 8/9
So, PE/QE = PF/RF
Hence, EF || QR.
[By Converse of basic proportionality theorem]
(iii) Given :
PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm, PF = 0.36 cm
Here, EQ = PQ - PE = 1.28 - 0.18 = 1.10 cm
And, FR = PR - PF = 2.56 - 0.36 = 2.20 cm
So, PE/EQ = 0.18/1.10 = 18/110 = 9/55
And, PE/FR = 0.36/2.20 = 36/220 = 9/55
∴ PE/EQ = PF/FR.
Hence, EF || QR.
[By Converse of basic proportionality theorem]
HOPE THIS ANSWER WILL HELP YOU...
PE = 3.9 cm,
EQ = 3 cm,
PF = 3.6 cm,
FR = 2.4 cm
Now we know,
Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides of the triangle, then the line divides these two sides proportionally.
So, if the lines EF and QR are to be parallel, then ratio PE:EQ should be proportional to PF:PR
e.g., PE/EQ = 3.9cm/3cm = 1.3
PF/FR = 3.6/2.4 = 3/2 = 1.5
hence, PE/EQ ≠ PF/FR
therefore EF is not parallel to QR .
similarly,
(ii) Given :
PE = 4cm
EQ =4.5cm
PF = 8cm
FR = 9cm
PE/EQ = 4/4.5 = 8/9
PF/FR = 8/9
here we see , PE/EQ = PF/FR
therefore, EF || QR
(iii) Given :
PQ = 1.28 cm
PR = 2.56 cm
PE = 1.8cm
PF = 3.6cm
PE/PQ = 1.8/3.6 = 1/2
PQ/PR = 1.28/2.56 = 1/2
we can see that PE/PQ = PQ/PR
therefore , EF || PQ
