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Basic Proportionality Theorem

If three para...

Question

If three parallel lines l, m, n which are cut by the transversals $¯ ¯¯¯¯¯¯ ¯ A B$ and $¯ ¯¯¯¯¯¯¯ ¯ C D$ at P, Q, R and E, F, G respectively, then which of the following is not true?

\frac{P Q}{Q R} = \frac{E F}{F G}

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\frac{P Q}{P R} = \frac{E F}{E G}

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\frac{Q R}{P R} = \frac{F G}{E G}

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\frac{P Q}{P R} = \frac{P E}{R G}

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Solution

The correct option is D $\frac{P Q}{P R} = \frac{P E}{R G}$

l || m || n are cut by the transversals $¯ ¯¯¯¯¯¯ ¯ A B$ and $¯ ¯¯¯¯¯¯¯ ¯ C D$ at P, Q, R and E, F, G respectively.
join PG which intersects m at O.

In $Δ P R G, Q O | | R G$
$\frac{P Q}{Q R} = \frac{P O}{O G}$ .........(i)
In $Δ P G E, O F | | P E$
$\frac{P O}{O G} = \frac{E F}{F G}$ .........(ii)
From (i) and (ii)
$\frac{P Q}{Q R} = \frac{E F}{F G}$ .........(iii)
$\Rightarrow \frac{P Q}{Q R} + 1 = \frac{E F}{F G} + 1$
$\Rightarrow \frac{P Q + Q R}{Q R} = \frac{E F + F G}{F G}$
$\Rightarrow \frac{P R}{Q R} = \frac{E G}{F G}$
$\Rightarrow \frac{Q R}{P R} = \frac{F G}{E G}$
Taking reciprocal of (iii), we get
$\frac{Q R}{P Q} = \frac{F G}{E F}$
$\Rightarrow \frac{Q R}{P Q} + 1 = \frac{F G}{E F} + 1$
$\Rightarrow \frac{P R}{P Q} = \frac{E G}{E F}$
$\Rightarrow \frac{P Q}{P R} = \frac{E F}{E G}$
Clearly, $\frac{P Q}{P R} \neq \frac{P E}{R G}$
Hence, the correct answer is option (4).

Suggest Corrections

Similar questions

State true or false:

In the following diagram, lines

l, m

and

n

are parallel to each other. Two transversals

p

and

q

intersect the parallel lines at points

A, B, C

and

P, Q, R

respectively as shown, then:

\frac{A B}{B C} = \frac{P Q}{Q R}

Q. If l, m, n are three parallel lines and the transversals

t_{1}

and

t_{2}

cut the lines l, m, n at the points A, B, C and P, Q, R as shown in the figure, then:

l, m

and

n

are three parallel lines intersected by transversals

p

and

q

such that

l, m

and

n

cut off equal intercepts

A B

and

B C

p

. Show that

l, m

and

n

cut off equal intercepts

D E

and

E F

q

also.

In the follow diagram, lines l, m and n are parallel to each other. Two transversals p and q intersect the parallel lines l, m and n at points A,B,C and P,Q,R respectively as shown below. Prove that $\frac{A B}{B C} = \frac{P Q}{Q R}$ .

Q. If

l, m, n

are three parallel lines and the transversal

t_{1}

and

t_{2}

cut the lines

l, m, n

at the points

A, B, C

and

P, O, R

as shown in the figure, then

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If three parallel lines l, m, n which are cut by the transversals ¯¯¯¯¯¯¯¯AB and ¯¯¯¯¯¯¯¯¯CD at P, Q, R and E, F, G respectively, then which of the following is not true?

If three parallel lines l, m, n which are cut by the transversals $¯ ¯¯¯¯¯¯ ¯ A B$ and $¯ ¯¯¯¯¯¯¯ ¯ C D$ at P, Q, R and E, F, G respectively, then which of the following is not true?