Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 42

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In a triangle $A B C,$ with usual notations the length of the bisector of internal angle $A$ is-

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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-2>Q 43

Let

f, g, h

be the lengths of the perpendiculars from the circum-centre of the

△ A B C

on the sides

B C, C A

and

A B

respectively. If

\frac{a}{f} + \frac{b}{g} + \frac{c}{h} = λ \frac{a b c}{f g h},

then the value of

^{'} λ^{'}

is-

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 44

In an acute-angled triangle $A B C, A P$ is the altitude. A circle drawn with $A P$ as its diameter cuts the sides $A B$ and $A C$ at $D$ and $E$ , respectively. Then the length $D E$ is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 45

A A_{1}, B B_{1}, C C_{1}

are the medians of a triangle

A B C

whose centroid is

G

. If the points

A, G, C_{1}

and

B_{1}

are concyclic, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 46

If

' l'

is the length of median from the vertex

A

to the side

B C

of a

△ A B C,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 47

The product of the distances of the incentre from the angular points of a

△ A B C

is-

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 48

In a triangle

A B C, B = 60 °

and

C = 45 ° .

Let

D

divides

B C

internally in the ratio

1 : 3,

then value of

\frac{\sin ∠ B A D}{\sin ∠ C A D}

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 49

In a triangle

A B C,

points

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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Alpha Question Bank for Engineering: Mathematics>Chapter 33 - Properties of Triangle>Exercise-2>Q 50

STATEMENT- $1$ : If $R$ be the circumradius of a $∆ A B C,$ then the circumradius of its excentral $Δ I_{1} I_{2} I_{3}$ is $2 R .$

STATEMENT- $2$ : If the circumradius of a triangle be $R,$ then the circumradius of its pedal triangle is $\frac{R}{2} .$