Phrases Properties Of Triangle Previous Year Question Paper and Solution with PDF

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Phrases Properties Of Triangle PYQ

Qus : 1

If $\Delta=a^2-(b-c)^2$ , where $\Delta$ is the are of the triangle ABC, then $tanA=$

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Qus : 2

In a triangle ABC, if the tangent of half the difference of two angles is equal to one third of the tangent of half the sum of the angles, then the ratio of the sides opposite to the angles is

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Qus : 3

In an acute-angled ΔABC the least value of secA+secB+secC is

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Qus : 4

If in a triangle ABC, the altitudes from the vertices A, B, C on opposite sides are in HP, then sin A, sin B, sin C are in

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Solution

If altitudes of a triangle are in HP then its side will be in AP because sides are inverse proportion to height as area is constant. a, b, c are sides of triangle.

a, b, c are in A.P.

sin A, sin B, sin C are in A.P.

Qus : 5

In a triangle, if the sum of two sides is x and their product is y such that (x+z)(x-z)=y, where z is the third side of the triangle , then triangle is

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Qus : 6

In a triangle ABC,

$a\cos ^2\frac{C}{2}+\, c\, \, {\cos }^2\frac{A}{2}=\frac{3b}{2}$ then the sides of the triangle are in

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Qus : 7

In a triangle ABC, let angle C = π/2. If R is the inradius and R is circumradius of the triangle ABC, then 2(r + R) equals

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Qus : 8

In a triangle ABC, angle A=90° and D is the midpoint of AC. What is the value of

equal to?

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Qus : 9

What is the largest area of an isosceles triangle with two edges of length 3?

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Qus : 10

If A, B and C is three angles of a ΔABC, whose area is Δ. Let a, b and c be the sides opposite to the angles A, B and C respectively. Is

$s=\frac{a+b+c}{2}=6$ , then the product

$\frac{1}{3} s^{2} (s-a)(s-b)(s-c)$ is equal to

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Qus : 11

The perimeter of a

$\Delta ABC$ is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

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Solution

Quick Solution

Given: Perimeter = 6 × Arithmetic Mean of sin A, sin B, sin C

Using Law of Sines: $\frac{a}{\sin A} = 2R$ , and a = 1 ⇒ $\sin A = \frac{1}{2R}$

Assume $A = 30^\circ \Rightarrow \sin A = \frac{1}{2} \Rightarrow 2R = 2$

⇒ b = 2 sin B, c = 2 sin C

Perimeter = $1 + b + c = 1 + 2 \sin B + 2 \sin C$

Mean = $\frac{\sin A + \sin B + \sin C}{3}$

Check: $1 + 2\sin B + 2\sin C = 6 \cdot \frac{1/2 + \sin B + \sin C}{3}$ ✅

✅ Final Answer: 30°

Qus : 12

In ΔABC, if a = 2, b = 4 and ∠C = 60°, then A and B are respectively equal to

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Qus : 13

If the angles of a triangle are in the ratio 2 : 3 : 7, then the ratio of the sides opposite to these angles is

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Qus : 14

A harbour lies in a direction 60° South of West from a fort and at a distance 30 km from it, a ship sets out from the harbour at noon and sails due East at 10 km an hour. The time at which the ship will be 70 km from the fort is

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