Geometry - Study Mode

[#186] There are 8 equidistant points A, B, C, D, E, F, G and H (in same order) on a circle. What is the value of ∠FDH (in degrees)?
Correct Answer

(B) 45

Explanation

Solution: $$eqalign{
& angle FOH = frac{{angle FOB}}{2} = {90^ circ } cr
& left[ { herefore FOB{ ext{ is a straight line}}}
ight] cr
& angle HDF = frac{{angle FOH}}{2} = frac{{{{90}^ circ }}}{2} = {45^ circ } cr} $$

[#187] If in any triangle, the angles are in the ratio of 1 : 2 : 1, then what will be the ratio of its sides?
Correct Answer

(D) 1 : $$sqrt 2 $$ : 1

Explanation

Solution: 1 : 2 : 1 → 4x = 180 x = 45 Ratio will be = $$x08oxed{1:sqrt 2 :1}$$

[#188] An isosceles ΔMNP is inscribed in a circle. If MN = MP = 16√5 cm, and NP = 32 cm, what is the radius (in cm) of the circle?
Correct Answer

(A) 20

Explanation

Solution: $$eqalign{
& { ext{Height of }}Delta { ext{MND}} cr
& = sqrt {{{left( {16sqrt 5 }
ight)}^2} - {{left( {frac{{32}}{2}}
ight)}^2}} cr
& = sqrt {{{left( {16sqrt 5 }
ight)}^2} - {{left( {16}
ight)}^2}} cr
& = 16sqrt {5 - 1} cr
& = 16 imes 2 cr
& = 32 cr
& { ext{R}} = frac{{{ ext{abc}}}}{{4Delta }} = frac{{16sqrt 5 imes 16sqrt 5 imes 32}}{{4 imes frac{1}{2} imes 32 imes 32}} = 20 cr} $$

[#189] In ΔABC, AD is a median and P is a point on AC such that AP : PD = 3 : 4. Then ar (ΔAPB) : ar (ΔABC) is equal to:
Correct Answer

(D) 3 : 14

Explanation

Solution: ΔAPB = 3 ΔABC = 7 + 7 = 14 (median divide into two equal part) ΔAPB : ΔABC = 3 : 14

[#190] ΔABC a right angled triangle has ∠B = 90° and AC is hypotenuse. D is its circumcenter and AB = 3 cms, BC = 4 cms. The value of BD is
Correct Answer

(C) 2.5 cms

Explanation

Solution: AC $$ = sqrt {{3^2} + {4^2}} = sqrt {9 + 16} $$ AC = 5 ∵ D is circumcenter, it is equidistant from all the center So AD = CD = BD = $$frac{5}{2}$$ = 2.5 cm