Geometry - Study Mode

[#221] If $$a$$ and $$b$$ are the lengths of the sides of a right angled triangle whose hypotenuse is 10 and whose area is 20, then the value of ($$a$$ + $$b$$) 2 is
Correct Answer

(C) 180

Explanation

Solution: In right ΔABC, a 2 + b 2 = 10 2 (by pt) . . . . (i) Area ΔABC = $$frac{1}{2}$$ ab = 20 ab = 40 (a + b) 2 = a 2 + b 2 + 2ab = 10 2 + 2(40) = 180

[#222] In the given figure, ABC is an equilateral triangle. Two circles of radius 4 cm and 12 cm are inscribed in the triangle. What is the side (in cm) of an equilateral triangle?
Correct Answer

(B) $$24sqrt 3 $$

Explanation

Solution: Now, In ΔAOF, ∠AFO = 90° ∠OAF = 30° ⇒ AF = $$4sqrt 3 $$ xa0cm ⇒ AO = 8 cm and AE = AO + OD + DO' + O'E = 8 + 4 + 12 + 12 = 36 cm ⇒ Median = 36 cm In equilateral Δ, $$eqalign{
& { ext{Median}} = frac{{sqrt 3 }}{2} imes { ext{Side}} cr
& { ext{36}} = frac{{sqrt 3 }}{2} imes { ext{Side}} cr
& { ext{Side}} = frac{{72}}{{sqrt 3 }} imes frac{{sqrt 3 }}{{sqrt 3 }} = 24sqrt 3 { ext{ cm}} cr} $$

[#223] ABCD is a square and ΔMAB is an equilateral triangle. MC and MD are joined. What is the degree measure of ∠MDC?
Correct Answer

(D) 75°

Explanation

Solution: ∠A = ∠B = ∠D = ∠C = 90° ΔAMB is equilateral Hence ∠A = 60° ∠DAM = 90° + 60° = 150° ∠ADM $$ = frac{{{{180}^ circ } - {{150}^ circ }}}{2} = frac{{{{30}^ circ }}}{2} = {15^ circ }$$ ∠MDC = 90° - 15° = 75°

[#224] Two chords of length $$a$$ unit and $$b$$ unit of a circle make angles 60° and 90° at the centre of a circle respectively, then the correct relation is:
Correct Answer

(A) $$b = sqrt 2 a$$

Explanation

Solution: ∠AOB = 60° ∠COD = 90° ⇒ length of chord AB = a ⇒ length of chord CD = b ⇒ AO = OB = AB = OD = OC = a ⇒ In ΔODC ⇒ OD 2 + OC 2 = CD 2 ⇒ a 2 + a 2 = b 2 ⇒ b = $$sqrt 2 $$ a

[#225] In a triangle ABC, P and Q are point on AB and AC, respectively, such that AP = 1 cm, PB = 3 cm, AQ = 1.5 cm and CQ = 4.5 cm. If the area off ΔAPQ is 12 cm 2 , then find the area of BPQC.
Correct Answer

(A) 192 cm 2