Two line segments AB and AC include an angle of $60^{0}$ where AB $= 5$ cm and AC $= 7$ cm. Locate points P and Q on AB and AC, respectively such that AP $= \frac{3}{4}$ AB and AQ $= \frac{1}{4}$ AC. Join P and Q and measure the length PQ.

3.25

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4.25

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5.25

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6.25

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Solution

The correct option is A $3.25$ cm
$G i v e n - ∠ B A C = 60^{O} A B = 5 c m, A C = 7 c m, a n d A P = \frac{3}{4} A B = a, A Q = \frac{1}{4} A C = b . T o f i n d o u t P Q = c = ? S o l u t i o n - A P = \frac{3}{4} A B = \frac{3}{4} \times 5 c m = 3.75 c m = a a n d A Q = \frac{1}{4} A C = A Q = \frac{1}{4} 7 c m = 1.75 c m = b a l s o ∠ B A C = 60^{O} = θ W e k n o w t h a t i f t w o s i d e s a & b o f a t r i a n g l e i n c l u d e a n a n g l e θ b e t w e e n t h e m t h e n t h e t h i r d s i d e c i s g i v e n b y c = \sqrt{a^{2} + b^{2} - 2 a b cos θ} ∴ P Q = c = \sqrt{a^{2} + b^{2} - 2 a b cos θ} ⟹ P Q = \sqrt{{3.75}^{2} + {1.75}^{2} - 2 \times 3.75 \times 1.75 \times cos 60^{O}} c m ⟹ P Q = \sqrt{{3.75}^{2} + {1.75}^{2} - 2 \times 3.75 \times 1.75 \times \frac{1}{2}} c m = \sqrt{10.5625} c m = 3.25 c m A n s - P Q = 3.25 c m$

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Similar questions

Question 1
Two line segments AB and AC include an angle of $60^{\circ}$ , where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that $A P = \frac{3}{4} A B$ and $A Q = \frac{1}{4} A C$ .Join P and Q measure the length PQ.
Thinking Process
(i) Firstly we find the ratio of AB in which P divides, it with the help of the relation $A P = \frac{3}{4} A B$
(ii) Secondly, we find the ratio of AC in which Q divides. It with the help of the relation $A Q = \frac{1}{4} A C$ .
(iii) Now, construct the line segment AB and AC in which P and Q respectively divide it in the ratio from step (i) and (ii) respectively.
(iv) Finally get the point P and Q. After that join PQ and get the required measurement of PQ.

Question 1
Two line segments AB and AC include an angle of $60^{\circ}$ , where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that $A P = \frac{3}{4} A B$ and $A Q = \frac{1}{4} A C$ .Join P and Q measure the length PQ.
Thinking Process
(i) Firstly we find the ratio of AB in which P divides, it with the help of the relation $A P = \frac{3}{4} A B$
(ii) Secondly, we find the ratio of AC in which Q divides. It with the help of the relation $A Q = \frac{1}{4} A C$ .
(iii) Now, construct the line segment AB and AC in which P and Q respectively divide it in the ratio from step (i) and (ii) respectively.
(iv) Finally get the point P and Q. After that join PQ and get the required measurement of PQ.