Maths · Trigonometrical identities and trigonometrical functions
Two line segments and include an angle of where and Locate points and on and res
Two line segments \( A B \) and \( A C \) include an angle of \( 60^{0} \) where \( A B=5 \mathrm{cm} \) and \( A C \) \( =7 \mathrm{cm} . \) Locate points \( \mathrm{P} \) and \( \mathrm{Q} \) on \( \mathrm{AB} \) and \( A C, \) respectively such that \( A P=\frac{3}{4} \) \( A B \) and \( A Q=\frac{1}{4} A C . \) Join \( P \) and \( Q \) and measure the length PQ.
- A. \( 3.25 \mathrm{cm} \)
- B. \( 4.25 \mathrm{cm} \)
- C. 5 .25 \mathrm{cm} \)
- D. \( 6.25 \mathrm{cm} \)
Step-by-step solution
Given AB=5 cm, AC=7 cm, ∠BAC=60°. AP = (3/4)AB = 3.75 cm, AQ = (1/4)AC = 1.75 cm. Using law of cosines in triangle APQ: PQ² = AP² + AQ² - 2·AP·AQ·cos60° = 3.75² + 1.75² - 2·3.75·1.75·0.5 = 14.0625 + 3.0625 - 6.5625 = 10.5625, so PQ = √10.5625 = 3.25 cm.