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Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Fill in the Blanks

Q. 1. In a ΔABC, ÐA = 90° and AD is an altitude. Complete the relation

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. BC

Solution. We know that altitude from right vertex to hypotenuse in right angled triangle divides it into two triangles each being similar to the original triangle.

ΔBDA - ΔBAC

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 2. ABC is a triangle, P is a point on AB, and Q is point on AC such that ∠AQP = ∠ABC. Complete the relation

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. AP2

Solution. In ΔAPQ  and  ΔACB
∠A = ∠A  (common)
∠AQP = ∠ABC  (given)
∴ ΔAPQ -  ΔACB  fig is  with (iii) (by AA similarity)  

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 3. ABC is a triangle with ∠B greater than ∠C. D and E are points on BC such that AD is perpendicular to BC and AE is the bisector of angle A . Complete the relation

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. ∠B

Solution. We have   ∠BAE  =  ∠CAE (given) and          
∠ADB = ∠ADC = 90° (given)

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Now ∠DAE = ∠BAE – ∠BAD
= ∠CAE – (90° – ÐB)
= (∠CAD – ∠DAE) – 90° + ∠B
= (90° – ∠C) – ∠DAE – 90° + ∠B

⇒ 2 ∠DAE = ∠B – ∠C Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 4. The set of all real number s a such that a2 + 2a, 2a + 3 and a2 + 3a + 8 are the sides of a triangle is ..................

Ans. (5, ∞)

Solution. If  a2 + 2a,  2a + 3, a2 + 3a + 8 are sides of a Δ then sum of any two sides is greater than the third side.
Let x = a2 + 2a   ;   y = 2a + 3  ;  z = a2 + 3a + 8
Then x + y > z ⇒ a2 + 4a + 3 > a2 + 3a + 8
⇒ a > 5     .... (1)
y + z > x
⇒ a2 + 5a + 11 > a2 + 2a ⇒  3a > – 11
⇒  a > – 11/3      .... (2)
z + x > y
⇒ 2a2 + 5a + 8 > 2a + 3  ⇒ 2a2 + 3a + 5 > 0
Here coeff. of a2 > 0 and D = 9 – 40 = –ve
∴ it is true for all values of a. Therefore, identity.
Combining (1) and (2), we get a > 5

∴ a ∈ (5,∞)


Q. 5. In a triangle ABC, if cot A, cot B, cot C ar e in A.P., then a2, b2, c2, are in .................. progression.

Ans. arithmetic

Solution. cot A,  cot B,  cot C are in A.P.
⇒ cot B – cot A = cot C – cot B

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ sin (A – B) sin (A + B) = sin (B + C) sin (B – C)
⇒ sin2A – sin2 B = sin2 B – sin2 C
⇒ a2 – b2 = b2 – c2  ⇒ a2, b2, c2 are in A.P..


Q. 6. A polygon of nine sides, each of length 2, is inscribed in a circle. The radius of the circle is ..................

Ans. Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Let AB = 2 units be one of the sides of the polygon.

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Then ∠AOB = 2p/9 where O is the centre of circle.
If OL ⊥ AB, then AL = 1 and  ∠AOL = π/9
∴ Radius of the circle
= OA = AL cosec π/9 = cosec π/9.


Q. 7. If the angles of a triangle are 30° and 45° and the included side is (√3 + 1) cms, then the area of the triangle is ..................

Ans. Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution.

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 8. If in a triangle ABC, Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEthen the value of the angle A is .................. degrees.

Ans. 90°

Solution. In ΔABC, we have

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 9. In a triangle ABC, AD is the altitude from A. Given b > c, ∠C = 23º and Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 113°

Solution.

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE


Q. 10. A circle is inscribed in an equilateral triangle of side a. The area of any square inscribed in this circle is ..................

Ans. Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Given that ABC is an equilateral Δ, of side a, r is the radius of circle inscribed in it

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

If  PQRS is the square inscribed in circle of radius r, then

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

∴ area of square = a2/6 sq. units.

Q. 11. In a triangle ABC, a : b : c = 4 : 5 : 6. The ratio of the radius of the circumcircle to that of the incircle is ..................

Ans. 16 : 7

Solution.

Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

The document Fill in the Blanks: Properties of Triangle | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Maths 35 Years JEE Main & Advanced Past year Papers.
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