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Q. 1. A triangle ABC has sides AB = AC = 5 cm and BC = 6 cm Triangle A'B'C' is the reflection of the triangle ABC in a line parallel to AB placed at a distance 2 cm from AB, outside the triangle ABC. Triangle A"B"C" is the reflection of the triangle A'B'C' in a line parallel to B'C' placed at a distance of 2 cm from B'C' outside the triangle A'B'C'. Find the distance between A and A".

Ans. Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Let L be the line parallel to side AB of ΔABC, at a distance of 2 cm from AB, in which the first reflection ΔA' B' C' is obtained.

Let L' be the second line parallel to B' C', at a distance of 2 cm from B' C', in which reflection of ΔA' B' C' is taken as ΔA"B"C".

In figure, size of ΔA"B"C" is same to the size of ΔA'B'C'.

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

From figure AA' = 4 cm and A'A" = 12 cm.  So to find AA" it suffices to know ∠AA'A", clearly

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ cos ∠AA'A" = cos (90° + α) = – sin α =  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and hence

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q. 2. (a) If a circle is inscribed in a right angled triangle ABC with the right angle at B, show that the diameter of the circle is equal to AB + BC – AC.
 (b) If a triangle is inscribed in a circle, then the product of any two sides of the triangle is equal to the product of the diameter and the perpendicular distance of the third side from the opposite vertex. Prove the above statement.

Solution. (a) Inradius of the circle is given by

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

2r = a + c – b ⇒ Diameter = BC + AB – AC

(b) Given a ΔABC in which AD ⊥ BC, AE is diameter of circumcircle of ΔABC.

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

To prove :
AB × AC = AE × AD
Construction : Join BE
Proof  : ABE = 90° (∠ in a semi circle)
Now in Δ's ABE and ADC

∠ABE = ∠ADC (each  90°)    
∠AEB = ∠ACD (∠’s in the same segment)
∴ ΔABE ~ ΔADC (by AA similarity)

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (Proved)


Q. 3. (a) A balloon is observed simultaneously from three points A, B and C on a straight road directly beneath it. The angular elevation at B is twice that at A and the angular elevation at C is thrice that at A. If the distance between A and B is a and the distance between B and C is b, find the height of the balloon in terms of a and b.
 (b) Find the area of the smaller part of a disc of radius 10 cm, cut off by a chord AB which subtends an angle of 
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEat the circumference.

Ans. Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. By exterior angle theorem, in the adjacent fig.

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∠APB = ∠BPC = α
Also in ΔABP,  ∠BAP  = ∠APB = a
⇒ AB = PB = a
Applying sine law in ΔPBC, we get

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Area of the segment APB = Area of the sector  AOB – area of ΔAOB

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 4. ABC is a triangle. D is the middle point of BC. If AD is perpendicular to AC, then prove that Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution.

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 5. ABC is a triangle with AB = AC. D is any point on the side BC. E and F are points on the side AB and AC, respectively, such that DE is parallel to AC, and DF is parallel to AB.

Prove that 

DF + FA + AE + ED = AB + AC

Solution. Given that AB = AC

∴ ∠1 = ∠2 ....(1)

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

But AB || DF (given) and BC is transversal

∴ ∠1 = ∠3        ....(2)

From equations (1) and (2)
∠2 = ∠3
⇒ DF = CF         ....(3)
Similaraly we can prove DE = BE

Now, DF  + FA + AE + ED = CF + FA + AE + BE    
= AC + AB    [using equation (3) and (4)]


Q. 6. (i) PQ is a vertical tower. P is the foot and Q is the top of the tower. A, B, C are three points in the horizontal plane through P. The angles of elevation of Q from A, B, C are equal, and each is equal to θ. The sides of the triangle ABC are a, b, c; and the area of the triangle ABC is D. Show that the height of the tower is Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
(ii) AB is a vertical pole. The end A is on the level ground. C is the middle point of AB. P is a point on the level ground. The portion CB subtends an angle β at P. If

AP = n AB, then show that Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. (i)  Let h be the height of tower PQ.

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Similarly in Δ's BPQ and CPQ we obtain
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ P is the circumcentre of

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

ΔABC with circum radius as  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 7. Let the angles A, B, C of a triangle ABC be in A.P. and let  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE Find the angle A.

Solution. As the angles A, B, C of ΔABC are in AP
∴ Let A = x – d, B = x, C  = x + d
But A + B + C = 180° (∠Sum prop. of Δ)
∴ x – d + x + x + d = 180°
⇒  3x = 180° ⇒ x = 60° ∴ ∠B = 60°
Now by sine law in ΔABC, we have

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 8. A vertical pole stands at a point Q on a horizontal ground. A and B are points on the ground, d meters apart. The pole subtends angles α and β at A and B respectively. AB subtends an angle γ at Q. Find the height of the pole.

Solution. Let ht. of pole PQ be h.

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

In ΔAPQ,   Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒    Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE    ....(1)

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       ....(2)
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
∴ – 2h2 cot α cot β cos γ + h2  cot2 α + h2 cot2 β = d2 

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 9. Four ships A, B, C and D are at sea in the following relative positions : B is on the straight line segment AC, B is due North of D and D is due west of C. The distance between B and D is 2 km. ∠BDA = 40°, ∠BCD = 25°. What is the distance between A and D? [Take sin25° = 0.423]

Solution. According to question figure is as follows

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Here, ∠BDC = 90°, BD = 2 km
∠BDA = 40°,   ⇒ ∠ADC = 130°,
∴ ∠DAC = 180° – (25° +130°) = 25°
From the figure, in Δ ABD, using sine law

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 10. The ex-radii r1, r2, r3 of  Δ ABC are in H.P. Show that its sides a, b, c are in A.P.

Solution. KEY CONCEPT :

Ex-radii of a DABC are  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  are in HP.  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE are in AP

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ s – a, s – b, s – c are in AP
⇒ – a, – b, – c are in AP.
⇒ a, b, c are in A.P.


Q. 11. For a triangle ABC it is given that cosA +  cosB + cosC= 3/2. Prove that the triangle is equilateral.

Solution. Given that CosA +  cosB + cosC= 3/2. in ΔABC

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Which is possible only when

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

From (1) and (2) A = B = 60° Thus we get A = B = C = 60°
∴ ΔABC is an equilateral Δ.


Q. 12. With usual notation, if in a triangle ABC;  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE then prove that  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Given that, in Δ ABC,

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

where a, b, c are the lengths of sides  BC, CA and AB respectively.

Let Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ b + c = 11 k     ....(1)
c + a = 12 k         ....(2)
a + b = 13 k        ....(3)

Adding the above three eqs. we get    

2 (a + b + c) = 36 k
⇒ a + b + c = 18 k ....(4)

Solving each of (1), (2) and (3) with (4), we get

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 13. A ladder rests against a wall at an angle a to the horizontal. Its foot  is pulled away from the wall through a distance a, so that it slides a distance b down the wall making an angle b with the horizon tal. Show that Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Let ℓ be the length of the ladder, then

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
In ΔOQB,   Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒  Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   ....(1)

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒  OA = ℓ cos α ....(2)    
Now a = OB – OA = ℓ (cos β – cos α) ....(3)
Also from ΔOAP,   OP = ℓ sin α
and in OQB  ;  OQ = ℓ sin β
∴ b = OP – OQ = ℓ (sin α – sin β) ....(4)
Dividing eq. (3)   by   (4)   we get

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 14. In a triangle ABC, t he median to th e side BC is of length Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEand it divides the angle A into angles 30° and 45°. Find the length of the side BC.

Solution. Let AD be the median in ΔABC.
Let ∠B = q then ∠C = 105°– θ
In ΔABD, using sine law, we get

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

In ΔACD, using sine law, we get

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Q. 15. If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, Show that a : b : c = 1 : 1 : √2

Solution. We are given that in ΔABC cos A cos B + sin A sin B sin C = 1
⇒ sin A sin B sin C = 1 – cos A cos B

⇒ Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   [∴sin C < 1]

⇒ 1 - cos A cos B < sin A sinB

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
But we know cos (A – B) < 1
∴ We must have cos  (A – B) = 1
⇒ A –  B = 0
⇒ A = B
∴ cosA cos A + sin A sin A sin C = 1   [For A = B]
⇒ cos2 A + sin2 A sin C = 1
⇒ sin2 A sin C = 1 – cos2 A
⇒ sin2 A sin C = sin2 A ⇒ sin2 A (sin C – 1) = 0
⇒ sin A = 0  or  sin  C = 1
The only possibility is sin C = 1 ⇒ C = π/2
∴ A + B = π/2
But  A = B ⇒ A = B = π/4
∴ By Sine law in ΔABC,

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence proved the result.


Q. 16. A sign-post in the form of an isosceles triangle ABC is mounted on a pole of height h fixed to the ground. The base BC of the triangle is parallel to the ground. A man standing on the ground at a distance d from the sign-post finds that the top vertex A of the triangle subtends an angle β and either of the other two vertices subtends the same angle α at his feet. Find the area of the triangle.

Solution. Let ABC be the isosceles triangular sign board with BC horizontal. DE be the pole of height h. Let the man be standing at P such that PE = d

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Also ATQ, ∠APE = β
∠CPF = α
Let AD = x be altitude of ΔABC.
As ΔABC is isosceles with AB = AC
∴ D is mid pt of  BC.
Hence BC = 2y.
Now in ΔAPE,

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE      ....(1)

Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE   ....(2)
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Properties of Triangle - 1 | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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