You visited us 1 times! Enjoying our articles? Unlock Full Access!

Solution of Triangle

Let ABC be a ...

Question

Let $A B C$ be a triangle with $G_{1}, G_{2}, G_{3}$ as the mid-points of $B C (a), A C (b)$ and $A B (c)$ respectively. Also, let $M$ be the centroid of the triangle. It is given that the circumcircle of $Δ M A C$ touches the side $A B$ of the triangle at point $A$ , then

\frac{A G_{1}}{b} = \frac{\sqrt{3}}{2}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

max {sin ∠ C A M + sin ∠ C B M} = \frac{2}{\sqrt{3}}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{a^{2} + b^{2}}{c^{2}} = 2

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

\frac{c^{2}}{a b} = \sqrt{2}

sin ∠ C A M + sin ∠ C B M

is maximum

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct option is D $\frac{c^{2}}{a b} = \sqrt{2}$ if $sin ∠ C A M + sin ∠ C B M$ is maximum
$(G_{3} A)^{2} = (G_{3} M) (G_{3} C)$
$∴ {(\frac{c}{2})}^{2} = \frac{1}{3} (G_{3} C)^{2}$
$∴ \frac{c^{2}}{4} = \frac{1}{3} (\frac{2 b^{2} + 2 a^{2} - c^{2}}{4}) \Rightarrow a^{2} + b^{2} = 2 c^{2} \dots (1)$

Now, in $Δ A G_{1} C$
$sin ∠ C A M = \frac{a sin C}{2 (A G_{1})}$
and in $Δ B C G_{2}$
$sin ∠ C B M = \frac{b sin C}{2 (B G_{2})}$
Also, $A G_{1} = \sqrt{\frac{2 b^{2} + 2 c^{2} - a^{2}}{4}} = \frac{\sqrt{3}}{2} b$ [Using Eq. $(1)$ ]
and $B G_{2} = \sqrt{\frac{2 a^{2} + 2 c^{2} - b^{2}}{4}} = \frac{\sqrt{3}}{2} a$
Now, $sin ∠ C A M + sin ∠ C B M = \frac{2 Δ}{\sqrt{3}} (\frac{a^{2} + b^{2}}{a^{2} b^{2}})$
$= \frac{1}{\sqrt{3}} (\frac{a^{2} + b^{2}}{a b}) sin C \dots (2)$
Also, $cos C = \frac{a^{2} + b^{2} - c^{2}}{2 a b} \Rightarrow a^{2} + b^{2} = 4 a b cos C$
So, from Eq. $(2)$
$sin ∠ C A M + sin ∠ C B M = \frac{2}{\sqrt{3}} sin 2 C \leq \frac{2}{\sqrt{3}}$
$(where C = \frac{π}{4})$

Suggest Corrections

Similar questions

Q. Let

A B C

be a triangle with

G_{1}, G_{2}, G_{3}

as the mid-points of

B C (a), A C (b)

and

A B (c)

respectively. Also, let

M

be the centroid of the triangle. It is given that the circumcircle of

Δ M A C

touches side

A B

of the triangle at point

A

, then

Q. Let the centroid of an equilateral triangle

A B C

be at the origin. Let one of the sides of the equilateral triangle be along the straight line

x + y = 3

. If

R

and

r

be the radius of circumcircle and incircle respectively of

△ A B C

, then

(R + r)

is equal to

Q. Let

A B C

be an equilateral triangle with side length

a

. Let

R

and

r

denote the radii of the circumcircle and the incircle of the triangle

A B C

respectively. Then, as a function of

a

, the ratio

\frac{R}{r}

Q. In a triangle

A B C

, let

G

denote its centroid and let

M, N

be points in the interiors of the segments

A B, A C

, respectively, such that

M, G, N

are collinear. If

r

denotes the ratio of the area of triangle

A M N

to the area of

A B C

then

Let $Δ$ $A B C$ be an equilateral triangle. Let P be a point on minor arc AB of its circumcircle . Which of the following holds :

Join BYJU'S Learning Program

Let ABC be a triangle with G1,G2,G3 as the mid-points of BC(a),AC(b) and AB(c) respectively. Also, let M be the centroid of the triangle. It is given that the circumcircle of ΔMAC touches the side AB of the triangle at point A, then

Let $A B C$ be a triangle with $G_{1}, G_{2}, G_{3}$ as the mid-points of $B C (a), A C (b)$ and $A B (c)$ respectively. Also, let $M$ be the centroid of the triangle. It is given that the circumcircle of $Δ M A C$ touches the side $A B$ of the triangle at point $A$ , then