Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 12

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In a triangle $X Y Z,$ let $x, y, z$ be the lengths of sides opposite to the angles $X, Y, Z$ , respectively, and $2 s = x + y + z$ . If

$\frac{s - x}{4} = \frac{s - y}{3} = \frac{s - z}{2}$ and area of the incircle of the triangle $X Y Z$ is $\frac{8 π}{3}$ , then

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Important Questions on Properties of Triangle

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 13

In a triangle

P Q R,

let

∠ P Q R = 30 °

and the sides

P Q

and

Q R

have lengths

10 \sqrt{3} a n d 10

units, respectively. Then, which of the following statement(s) is (are) TRUE?

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 14

If in a triangle

A B C,

a \cos^{2} (\frac{C}{2}) + c \cos^{2} (\frac{A}{2}) = \frac{3 b}{2},

then the sides

a, b

and

c

-

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 15

In a triangle

A B C

, medians

A D

and

B E

are drawn. If

A D = 4, ∠ D A B = \frac{π}{6}

and

∠ A B E = \frac{π}{3}

, then the area of

Δ A B C

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 16

The sides of a triangle are

\sin α, \cos α

and

\sqrt{1 + \sin α \cos α}

for some

0 < α < \frac{π}{2} .

Then, the greatest angle of the triangle is

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 17

In triangle

A B C,

let

∠ C = \frac{π}{2}

. If

r

is the inradius and

R

is circumradius of the triangle, then

2 (r + R)

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 18

If in a

Δ A B C,

the altitudes from the vertices

A, B

and

C

on opposite sides are in

H P,

then

\sin A, \sin B

and

\sin C

are in

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 19

For a regular polygon, let

r

and

R

be the radii of the inscribed and the circumscribed circles. A false statement among the following is

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-3>Q 20

With the usual notation in

∆ A B C

, if

∠ A + ∠ B = 120 °, a = \sqrt{3} + 1

units and

b = \sqrt{3} - 1

units, then the ratio

∠ A : ∠ B

is