In $$\triangle ABC$$​,$$\overline{PQ} \parallel \overline{AB}$$​. P and Q are on BC and CA respectively. If CQ : QA = 1 : 3 and CP = 4, then what is the value of BC?

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Correct option is A

Given:

In △ABC:

PQ∥ABP (line PQ is parallel to AB), and P and Q are points on BC and CA, respectively.

The ratio CQ : QA = 1 : 3

CP = 4

Concept Used:

If a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides the two sides in the same ratio.

Solution:

Since PQ∥AB, by the Basic Proportionality Theorem (also known as Thales' theorem), the line PQ divides the sides BC and CA in the same ratio. 

From the given ratio CQ : QA = 1 : 3,

CQ =$$\frac{1}{4}CA$$​,

QA = $$\frac{3}{4}CA$$​​

Let the total length of CA be x. Then:

​$$CQ = \frac{1}{4}x, \quad QA = \frac{3}{4}x$$​​

Similarly, PQ∥AB, the point P divides BC in the same ratio 1 : 3 (as CQ : QA = 1 : 3):

CP : PB = 1 : 3

CP = 4, we can express PB as:

PB = 3×CP = 3×4 = 12

Thus, the total length of BC is:

BC = CP + PB = 4 + 12 = 16

The value of BC is 16 cm