Pedal Triangle

Q. Relation between circumradius, side lengths and area of the triangle can be given by $R=\frac{abc}{4\Delta }$ where a, b, c represent the sides and $\Delta$ represents the area

1. True
2. False

Q. Prove that cos30^°-sin20^°/cos40^°+cos20^°=4/rt3.cos40^°.cos80^°

Q. If in a triangle $ABC,$ $AB=5$ units, $\angle B={\cos }^{-1}\left(\frac{3}{5}\right)$ and radius of circumcircle of $△ABC$ is $5$ units, then the area (in sq. units) of $△ABC$ is

1. $10+6\sqrt{2}$
2. $6+8\sqrt{3}$
3. $8+2\sqrt{2}$
4. $4+2\sqrt{3}$

Q. For any triangle ABC prove sin(B-C)/sin(B+C)=b²-c²/a².

Q. In a triangle $ABC,$ the minimum value if the sum of the squares of sides is ($\Delta$ is the area of the triangle $ABC$)

1. $3\sqrt{3}\Delta$
2. $5\sqrt{3}\Delta$
3. $4\sqrt{3}\Delta$
4. $2\sqrt{3}\Delta$

Q. The area of an acute triangle $ABC$ is $\Delta$, the area of its pedal triangle is '$p$', where $\cos B=\frac{2p}{\Delta }$ and $\sin B=\frac{2\sqrt{3}p}{\Delta }$. The value of $8({\cos }^{2}A\cos B+{\cos }^{2}C)$ is

1. $1$
2. $2$
3. $3$
4. $4$

Q. In $△ABC,8\Delta =(b+c)(bc+1)$, then circumradius of $△ABC$ is ($\Delta$ denotes area of triangle and $a,b,c$ are length of sides $BC,AC$ and $AB$ respectively)

1. $\sqrt{\Delta }$
2. $\frac{1}{\sqrt{2\Delta }}$
3. $\sqrt{2\Delta }$
4. $\frac{1}{\sqrt{\Delta }}$

Q. In an acute angled triangle ABC, D, E and F are the feets of perpendiculars from A, B and C respectively and H is the orthocenter.
$\mathrm{If}\sin A=\frac{3}{5}\mathrm{and}\mathrm{BC}=39,\mathrm{then}\mathrm{length}\mathrm{of}\mathrm{AH}=$
.

1. 52
2. 65
3. 26
4. 39

Q. In a triangle $ABC$, if $B=30°,C=45°$ and $a=\sqrt{3}+1$ , then find the area of triangle $ABC$.

Q. In an acute angled triangle ABC, D, E and F are the feets of perpendiculars from A, B and C respectively and H is the orthocenter.
$\mathrm{If}\sin A=\frac{3}{5}\mathrm{and}\mathrm{BC}=39,\mathrm{then}\mathrm{length}\mathrm{of}\mathrm{AH}=$
.

1. 52
2. 65
3. 26
4. 39