Solution of Triangle
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Q.
A statue, tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is and from the same point the angle of elevation of the top of the pedestal is . Find the height of the pedestal.
Q. Let O be the origin, and −−→OX, −−→OY, −−→OZ be three unit vectors in the directions of the sides −−→QR, −−→RP, −−→PQ, respectively, of a triangle PQR.
If the triangle PQR varies, then the minimum value of
cos(P+Q)+cos(Q+R)+cos(R+P)
If the triangle PQR varies, then the minimum value of
cos(P+Q)+cos(Q+R)+cos(R+P)
- −53
- 32
- 53
- −32
Q. Consider a triangular plot ABC with sides AB=7m, BC=5m and CA=6m. A vertical lamp-post at the mid point D of AC subtends an angle 30∘ at B. The height (in m) of the lamp-post is :
- 32√21
- 7√3
- 2√21
- 23√21
Q.
What are the formulae for and .
Q.
If the median of ΔABC through A is perpendicular to AB then
tanA+tanB=0
None of these
2tanA+tanB=0
tanA+2tanB=0
Q. In a ΔABC, let a, b and c denote the length of sides opposite to vertices A, B and C respectively. If b=2, c=√3 and ∠BAC=π6, then value of circumradius of triangle ABC is
Q. The lengths of the sides of a triangle (in units) ABC are AB=10, BC=7, CA=√37 then length of the median through vertex C is
- 3√2 units
- 2√3 units
- 3√3 units
- 4√2 units
Q.
Given , find in degrees. Round the value to the nearest hundredth.
Q. If AD, BE and CF are the medians of ΔABC, then the value of (AD2+BE2+CF2):(BC2+CA2+AB2) is:
- 34
- 32
- 14
- 12
Q.
If the median AD of a triangle ABC is perpendicular to AB then the value of tan A+ 2tan B
None of these
0
1
3
Q. If P is a 3×3 orthogonal matrix α, β, γ are the angles made by a straight line OX, OY, OZ and A=⎡⎢⎣sin2αsinαsinβsinαsinγsinαsinβsin2βsinβsinγsinαsinγsinβsinγsin2γ⎤⎥⎦&Q=PTAP.
If PQ6PT=KA then k is
If PQ6PT=KA then k is
Q. In the given figure, ΔABC is a right angled triangle at A. The hypotenuse is divided into n equal parts. Let PQ be the segment that contains the midpoint of the hypotenuse.
If a and h be the length of the hypotenuse and altitude to the hypotenuse of the triangle respectively. Then which of the following option is correct ?
(Given that n is an odd integer and α=∠PAQ)
If a and h be the length of the hypotenuse and altitude to the hypotenuse of the triangle respectively. Then which of the following option is correct ?
(Given that n is an odd integer and α=∠PAQ)
- cotα=4nh(n2−1)a
- tanα=4an(n2−1)h
- cotα=4an(n2−1)h
- tanα=4nh(n2−1)a
Q. Let AD be a median of the △ABC. If AE and AF are medians of the triangle ABD and ADC, respectively, and AD=m1, AE=m2, AF=m3, then a28 is equal to
- m21+m23−2m22
- m22+m23−2m21
- m21+m22−2m23
- m21+m22−2m23
Q. Which of the following expressions have value equal to four times the area of the triangle ABC?
(All symbols used have their usual meaning in a triangle)
(All symbols used have their usual meaning in a triangle)
- rs+r1(s−a)+r2(s−b)+r3(s−c)
- (a+b+c)2cotA2+cotB2+cotC2
- (a2+b2−c2)tanB
- b2sin2C+c2sin2B
Q. In a triangle ABC, points X and Y are on AB and AC, respectively, such that XY is parallel to BC. Which of the following two equalities always hold? (Here [PQR] denotes the area of triangle PQR.)
(I) [BCX]=[BCY]
(II) [ACX]⋅[ABY]=[AXY]⋅[ABC]
(I) [BCX]=[BCY]
(II) [ACX]⋅[ABY]=[AXY]⋅[ABC]
- Neither (I) nor (II)
- (II) only
- (I) only
- Both (I) and (II)
Q.
If AD, BE and CF are the medians of a △ABC then (AD2+BE2+CF2):(BC2+CA2+AB2) is equal to
4:3
3:2
3:4
2:3
Q. Let ABC be a triangle with G1, G2, G3 as the mid-points of BC(a), AC(b) and AB(c) respectively. Also, let M be the centroid of the triangle. It is given that the circumcircle of ΔMAC touches the side AB of the triangle at point A, then
- AG1b=√32
- max{sin∠CAM+sin∠CBM}=2√3
- a2+b2c2=2
- c2ab=√2 if sin∠CAM+sin∠CBM is maximum
Q. If b > c sin B, b < c and B is acute angle then number of triangles possible following the given conditions is 1.
- True
- False
Q. Let P be a point inside a triangle ABC with ∠ABC=90∘. Let P1 and P2 be the images of P under reflection in AB and BC respectively. The distance between the circumcenters of triangles ABC and P1PP2 is
- AB2
- AP+BP+CP3
- AC2
- AB+BC+AC2
Q. Which of the following expressions have value equal to four times the area of the triangle ABC?
(All symbols used have their usual meaning in a triangle)
(All symbols used have their usual meaning in a triangle)
- rs+r1(s−a)+r2(s−b)+r3(s−c)
- (a+b+c)2cotA2+cotB2+cotC2
- (a2+b2−c2)tanB
- b2sin2C+c2sin2B
Q. Match the following by appropriately matching the lists based on the information given in Column I and Column II.
Column IColumn II (Typeof△ABC)a.cotA2=b+ca p. always right angled b. atanA+btanB=(a+b)tanA+B2 q. always isosceles c. acosA=bcosB r. may be right angled d. cosA=sinB2sinC s. may be right angled isosceles
Column IColumn II (Typeof△ABC)a.cotA2=b+ca p. always right angled b. atanA+btanB=(a+b)tanA+B2 q. always isosceles c. acosA=bcosB r. may be right angled d. cosA=sinB2sinC s. may be right angled isosceles
- a−p, q; b−q, r; c−p, s; d−q, r
- a−p, q; b−q, s; c−p, s; d−q, s
- a−p, s; b−q, r, s; c−r, s; d−q, r, s
- a−p, s; b−q, r; c−p, s; d−q, r
Q. Let ABC be a triangle with incentre I and r. Let D, E, F be the feet of the perpendiculars from I to the sides BC, CA and AB respectively, If r1, r2, andr3 are the radii of circles inscribed in the quadrilaterals AFIE, BDIF and CEID respectively, prove that
r1r−r1+r2r−r2+r3r−r3=r1r2r3(r−r1)(r−r2)(r−r3)
r1r−r1+r2r−r2+r3r−r3=r1r2r3(r−r1)(r−r2)(r−r3)
- True
- False
Q. The top of a ladder reaches a point on the wall 5m above the ground. If the foot of the ladder makes an angle of 30∘ with the ground, find the length of the ladder.
Q. In Δ ABC having vertices A(a cosθ1, a sinθ1), B(a cosθ2, a sinθ2) and C(a cosθ3, a sinθ3) is equilateral, then which of the followings is/are true?
- cosθ1+cosθ2+cosθ3=0
- sinθ1+sinθ2+sinθ3=0
- cos(θ1−θ2)+cos(θ2−θ3)+cos(θ3−θ1)=−32
- ∣∣sin(θ1−θ22)∣∣=sin∣∣(θ2−θ32)∣∣=∣∣sin(θ1−θ32)∣∣
Q. In a triangle ABC, r1, r2, r3 are the ex-radii of the ex-circles. If a=r1+r2+r3r and b=r1r2r3r3, where r is the inradius then the minimum value of ((amin3)tan2A+(bmin9)cot2A) is
Q. Let ABC be a triangle with G1, G2, G3 as the mid-points of BC(a), AC(b) and AB(c) respectively. Also, let M be the centroid of the triangle. It is given that the circumcircle of ΔMAC touches side AB of the triangle at point A, then
- AG1b=√32
- max{sin∠CAM+sin∠CBM}=2√3
- a2+b2c2=2
- c2ab=√2 if sin∠CAM+sin∠CBM is maximum
Q. In a right angled triangle, altitude divides the hypotenuse into two segments of length 8 units and 18 units, then the length of the smallest side is
(correct answer + 1, wrong answer - 0.25)
(correct answer + 1, wrong answer - 0.25)
- 8√2 units
- 6√13 units
- 4√13 units
- 8√3 units
Q.
Given angle A=60∘, c=√3−1, b=√3+1. Solve the triangle
- a=√6, B=15∘, C=100∘
- a=√12, B=15∘, C=105∘
- a=√6, B=105∘, C=15∘
- a=√6, B=15∘, C=105∘
Q. Given angle A=60∘, c=√3−1, b=√3+1. Solve the triangle
- a=√6, B=15∘, C=100∘
- a=√6, B=15∘, C=105∘
- a=√12, B=15∘, C=105∘
- a=√6, B=105∘, C=15∘
Q. Number of triangles possible for a given b, c and B(acute angle) under the condition that b < c sin B. Where b, c are the sides and B is the angle opposite to b.
- 2
- 3
- 0
- 1