Geometry - Study Mode
[#86] Two circles C 1 and C 2 touch each other internally at P. Two lines PCA and PDB meet the circles C 1 in C, D and C 2 in A, B respectively. If ∠BDC = 120°, then the value of ∠ABP is equal to
Correct Answer
(A) 60°
Explanation
Solution: According to question Given: ∠BDC = 120°, ∠ABP =? ∴ ∠CDP = 180° - ∠BDC ∠CDP = 180° - 120° ∠CDP = 60° CD || AB ∴ ∠CDP = ∠ABP = 60°
[#87] In a ΔABC, points P, Q and R are taken on AB, BC and CA, respectively, such that BQ = PQ and QC = QR. If ∠BAC = 75°, what is the measure of ∠PQR (in degrees)?
Correct Answer
(B) 30
[#88] In the given figure, PQRS is a square of side 20 cm and SR is extended to point T. If the length of QT is 25 cm, then what is the distance (in cm) between the centers O 1 and O 2 of the two circles?
Correct Answer
(A) $$5sqrt {10} $$
Explanation
Solution: $$eqalign{
& R{T^2} = {25^2} - {20^2} cr
& RT = 15 cr
& { ext{Inradius }}left( { ext{r}}
ight) = frac{{20 + 15 - 25}}{2} = frac{{10}}{2} = 5 cr
& { ext{In }}Delta {O_1}A{O_2} cr
& {O_1}O_2^2 = {O_1}{A^2} + AO_2^2 cr
& = {left( 5
ight)^2} + {left( {15}
ight)^2} cr
& = 25 + 225 cr
& = 250 cr
& {O_1}{O_2} = 5sqrt {10} { ext{ cm}} cr} $$
[#89] ln ΔABC, ∠A = 66°. AB and AC are produced to points D and E, respectively. If the bisectors of ∠CBD and ∠BCE meet at the point O, then ∠BOC is equal to:
Correct Answer
(C) 57°
Explanation
Solution: $$eqalign{
& angle A = {66^ circ } cr
& angle BOC = ? cr
& angle BOC = {90^ circ } - frac{{angle A}}{2} cr
& = {90^ circ } - frac{{{{66}^ circ }}}{2} cr
& = {90^ circ } - {33^ circ } cr
& = {57^ circ } cr} $$
[#90] In a circle with centre O, AD is a diameter and AC is a chord. Point B is on AC such that OB = 7 cm and ∠OBA = 60°. If ∠DOC = 60°, then what is the length of BC?
Correct Answer
(C) 7 cm
Explanation
Solution: $$eqalign{
& OB = 7,{ ext{cm}} cr
& { ext{Sine Rule in }}Delta BOC cr
& frac{{OB}}{{sin {{30}^ circ }}} = frac{{BC}}{{sin {{30}^ circ }}} cr
& BC = 7,{ ext{cm}} cr} $$