Area Using Semiperimeter

Q.

How do you find the value of $\cos \frac{\mathrm{\pi }}{6}$?

Q.

In a $△ABC,$ if 2s = a + b + c and (s-b)(s-c)=x $si{n}^2\frac{A}{2},$ then x = [MP PET 1992]

1. ca

2. bc

3. ab

4. abc

Q. Is the set $H=\{t|t$ is a triangle having four sides$\}$ empty?

1. Yes
2. Data insufficient
   
3. Ambiguous
4. No

Q.

In acute angled triangle $ABC,r+r_1=r_2+r_3$ and $\angle B>\frac{\pi }{3}$ then

1. $b+3c<3a<3b+3c$

2. $b+2c<2a<2b+2c$

3. $b+4c<4a<2b+4c$

4. $b+4c<4a<4b+4c$

Q.

In acute angled triangle $ABC,r=r_2+r_3-r_1$ and $\angle B>\frac{\pi }{3}$ then exhaustive range of $\frac{a-c}{b}$ is

1. $(\frac{1}{2},1)$

2. $(\frac{1}{4},\frac{1}{2})$

3. $(\frac{1}{4},1)$

4. None of these

Q. The area of the triangle whose two sides are given by 4i - j + k and 3i + j - k is

1. $14/\sqrt{2}$
2. $7/\sqrt{2}$
3. $7\sqrt{2}$
4. $14\sqrt{2}$

Q. If sides of a triangle are 3, 5, 7 then the area of that triangle is -

1. $\sqrt{23.12}$
   
2. $\sqrt{42.18}$
   
3. $\sqrt{62.78}$
   
4. $\sqrt{97.89}$
   

Q. Find m if $m[a{\sin }^2\frac{C}{2}+c{\sin }^2\frac{A}{2}]=c+a-b$

Q. If sides of a triangle are 3, 5, 7 then the area of that triangle is -

1. $\sqrt{42.18}$
   
2. $\sqrt{62.78}$
   
3. $\sqrt{97.89}$
   
4. $\sqrt{23.12}$
   

Q. In a triangle $XYZ$, let $x,y,z,$ be the lengths of sides opposite to the angles $X,Y,Z,$ respectively, and $2s=x+y+z$. If $\frac{s-x}{4}=\frac{s-y}{3}=\frac{s-z}{2}$ and the area of the incircle of the triangle $XYZ$ is $\frac{8\pi }{3},$ then:

1. Area of the triangle $XYZ$ is $6\sqrt{6}$
2. The radius of the circumcircle of the triangle $XYZ$ is $\frac{35}{6}\sqrt{6}$
3. $\sin \frac{X}{2}\sin \frac{Y}{2}\sin \frac{Z}{2}=\frac{4}{35}$
4. ${\sin }^2\left(\frac{X+Y}{2}\right)=\frac{3}{5}$