TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 8

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Given an isosceles triangle with equal sides having length $b$ , base angle $α < \frac{π}{4} . R, r$ the radii and $O$ , I the centres of the circumcircle and incircle respectively. Then

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Important Questions on Properties of Triangles

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 9

In a triangle

A B C

, point

D

and

E

are taken on side

B C

such that

B D = D E = E C

. If angle

A D E =

angle

A E D = θ

, then

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 10

Which of the following holds good for any triangle

ABC, w h e r e a, b, c

are the lengths of the sides and

R

is circumradius,

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 11

If the vertices

P, Q, R

of a triangle

PQR

are rational points, which of the following points of the triangle

PQR

is/are always rational point(s)?

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 12

In a triangle

ABC

, the minimum value of

\cot^{2} A + \cot^{2} B + \cot^{2} C

is equal to

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 13

If

a

and

b

be the length of the sides and

c

the length of hypotenuse of a right-angled triangle, then

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 14

If in a right-angled triangle, right angled at $A, sinB$ and $cosB$ are rational numbers then

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 15

.In a triangle

ABC

, if

Z = |\begin{matrix} e^{2 i A} & e^{- i C} & e^{- i B} \\ e^{- i C} & e^{2 i B} & e^{- i A} \\ e^{- i B} & e^{- i A} & e^{2 i C} \end{matrix}|

than

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TRIGONOMETRY FOR JEE MAIN AND ADVANCED>Chapter 11 - Properties of Triangles>EXERCISE>Q 16

If

a, b, c

are the sides of a right-angled

∆ A B C

and

p, d \in R

such that

(a^{2} + b^{2} + c^{2}) p^{2} - 2 (a b + b c + c d) p + (c^{2} + d^{2}) \leq 0

, then cosine of the greatest acute angle of

∆ A B C

is