Geometry - Study Mode

[#206] In ΔABC, ∠C = 90°, point P and Q are on the sides AC and BC, respectively, such that AP : PC = BQ : QC = 1 : 2. Then, $$frac{{{ ext{A}}{{ ext{Q}}^2} + { ext{B}}{{ ext{P}}^2}}}{{{ ext{A}}{{ ext{B}}^2}}}$$ xa0 is equal to:
Correct Answer

(C) $$frac{{13}}{9}$$

Explanation

Solution: Let the side of BC & CA are 3x and 3y QC = 2x QB = x PC = 2y AP = y In triangle AQC AQ 2 = (3y) 2 + (2x) 2 = 9y 2 + 4x 2 BP 2 = (3x) 2 + (2y) 2 = 9x 2 + 4y 2 AB 2 = (3x) 2 + (3y) 2 = 9x 2 + 9y 2 = 9(x 2 + y 2 ) $${ ext{Now, }}frac{{{ ext{A}}{{ ext{Q}}^2} + { ext{B}}{{ ext{P}}^2}}}{{{ ext{A}}{{ ext{B}}^2}}} = frac{{13left( {{x^2} + {y^2}}
ight)}}{{9left( {{x^2} + {y^2}}
ight)}} = frac{{13}}{9}$$

[#207] In the given figure, PQRS is a cyclic quadrilateral. What is the measure of the angle PQR if PQ is parallel to SR?
Correct Answer

(B) 110°

Explanation

Solution: ∠P + ∠R = 180° (∵ PQRS is a cyclic quadrilateral) 110° + ∠R = 180° ∠R = 70° ∠R + ∠Q = 180° (supplementary angle) 70° + ∠Q = 180° ∠PQR = 110° Note:- In any cyclic quadrilateral if two sides are parallel, then non-parallel sides are also equal in length. This type of quadrilateral is called an isosceles trapezium. If AB || CD Then AD = BC and ∠A = ∠B, ∠C = ∠D

[#208] AB is a diameter of the circle with centre O, CD is chord of the circle, If ∠BOC = 120°, then the value of ∠ADC is ?
Correct Answer

(B) 30°

Explanation

Solution: From figure ∠BOC = 120 ∴ ∠AOC = 180 - 120 = 60 So, ∠ADC = $$frac{1}{2}$$∠AOC (Angle made on circumference is half of the angle made on centre) = $$frac{1}{2}$$ × 60 So, ∠ADC = 30°

[#209] PA and PB are two tangents from a point P outside the circle with centre O. If A and B are points on the circle such that ∠APB = 142°, then ∠OAB is equal to:
Correct Answer

(C) 71°

Explanation

Solution: $$angle { ext{OAB}} = frac{{{{180}^ circ } - {{38}^ circ }}}{2} = {71^ circ }$$

[#210] A circle is inscribed in the triangle ABC whose sides are given as AB = 10, BC = 8, CA = 12 units as shown in the figure. The value of AD × BF is:
Correct Answer

(B) 21 units