Distances between Special Points in a Triangle

Q. If ABC is an acute-angled triangle with circumcenter O and Orthocenter H. If AO = AH, then angle A = .

1. $\frac{\pi }{6}$
2. $\frac{\pi }{8}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{4}$

Q. In the following figure, the area of the pedal triangle is -

1. $\frac{1}{2}{R}^2.\text{sin A.sin B.sin C}$
2. $\frac{1}{2}{R}^2.\text{sin 2A.sin 2B.sin 2C}$
3. $\frac{1}{2}{R}^2.\text{cos A.cos B.cos C}$
4. None of these

Q. Let origin $O$ be the orthocentre of an equilateral triangle $ABC$. If $\overrightarrow{OA}=\overrightarrow{a},\overrightarrow{OB}=\overrightarrow{b},\overrightarrow{OC}=\overrightarrow{c},$ then $\overrightarrow{AB}+2\overrightarrow{BC}+3\overrightarrow{CA}=$

1. $3\overrightarrow{c}$
2. $3\overrightarrow{a}$
3. $\overrightarrow{0}$
4. $3\overrightarrow{b}$

Q. Let $AB$ be a line segment with midpoint $C$, and $D$ be the midpoint of $AC$. Let $C_1$ be the circle with diameter $AB$, and $C_2$ be the circle with diameter $AC$. Let $E$ be a point on $C_1$ such that $EC$ is perpendicular to $AB$. Let $F$ be a point on $C_2$ such that $DF$ is perpendicular to $AB$, and $E$ and $F$ lie on opposite sides of $AB$. Then the value of $\sin \angle FEC$ is

1. $\frac{1}{\sqrt{10}}$
2. $\frac{2}{\sqrt{10}}$
3. $\frac{2}{\sqrt{13}}$
4. $\frac{1}{\sqrt{13}}$

Q. Let in a $△ABC,$ $x,y,z$ are the lengths of perpendicular drawn from the vertices of the triangle to the opposite sides $a,b,c.$ Then the value of $\frac{c\sin B+b\sin C}{x}+\frac{a\sin C+c\sin A}{y}+\frac{b\sin A+a\sin B}{z}$ is equal to

Q. In the following figure, the area of the pedal triangle is -

1. $\frac{1}{2}{R}^2.\text{sin A.sin B.sin C}$
2. $\frac{1}{2}{R}^2.\text{sin 2A.sin 2B.sin 2C}$
3. $\frac{1}{2}{R}^2.\text{cos A.cos B.cos C}$
4. None of these

Q. In a right angled triangle PQR, Circumcenter and orthocenter are at a distance of ------ units from each other. (Where R is the circumradius)

1. R

2. 2R

3. 3R

4. 4R

Q.

In an equilateral triangle sum of the distances of orthocenter from all the vertices is equal to -
(Where 'R' is the circumradius of triangle)

1. R
   

2. 2R

3. 3R

4. 4R

Q. The points $D,E,F$ divide $\overrightarrow{BC},\overrightarrow{CA}$ and $\overrightarrow{AB}$ of the triangle $ABC$ in the ratio $1:4,3:2$ and $3:7$ respectively and the point $K$ divides $\overrightarrow{AB}$ in the ratio $1:3$, then $(\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF}):\overrightarrow{CK}$ is equal to

1. $1:1$
2. $2:5$
3. $5:2$
4. $3:5$

Q. If in an isosceles right angled triangle the area of circumcircle is 16 times the area of incircle, also all points circumcenter, orthocenter, incenter, centroid lie in the 1st quadrant, orthocenter being at origin. Find the distance between circumcenter and incenter.

1. 4R - 2 $\sqrt{2}$ r
2. 4r - 2 $\sqrt{2}$ r
3. 4R - 2 $\sqrt{2}$ R
4. 4 - 2 $\sqrt{2}$ r

Q. Let $r$ and $R$ be the inradius and the circumradius of a $△ABC$. Let $\theta$ be the angle between the line joining the incentre and the circumcentre of the $△ABC$ and $BC$. Then $\theta$ is equal to

1. ${\tan }^{-1}\left(\frac{\cos B+\cos C-1}{\sin C-\sin B}\right)$
2. ${\tan }^{-1}\left(\frac{\sin C-\sin B}{\cos B+\cos C-1}\right)$
3. ${\tan }^{-1}\left(\frac{r-R\cos A}{r\cot \frac{B}{2}-R\sin A}\right)$
4. ${\tan }^{-1}\left(\frac{r\cos A-R}{r\sin A-R\cot \frac{B}{2}}\right)$

Q. If $\overrightarrow{a},\overrightarrow{b},$ and $\overrightarrow{c}$ are unit vectors such that $\overrightarrow{a}+2\overrightarrow{b}+2\overrightarrow{c}=\overrightarrow{0},$ then $|\overrightarrow{a}\times \overrightarrow{c}|$ is equal to :

1. $\frac{\sqrt{15}}{4}$
2. $\frac{1}{4}$
3. $\frac{15}{16}$
4. $\frac{\sqrt{15}}{16}$

Q. $\underset{0}{\overset{1/2}{\int }}\frac{dx}{(1+x^2)\sqrt{1-x^2}}$ is equal to

1. $\frac{1}{\sqrt{2}}{\tan }^{-1}\left(\sqrt{\frac{2}{3}}\right)$
2. $\frac{2}{\sqrt{2}}{\tan }^{-1}\left(\frac{3}{\sqrt{2}}\right)$
3. $\frac{\sqrt{2}}{2}{\tan }^{-1}\left(\frac{3}{2}\right)$
4. $\frac{\sqrt{2}}{2}{\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

Q. Let $r$ and $R$ be the inradius and the circumradius of a $△ABC$. Let $\theta$ be the angle between the line joining the incentre and the circumcentre of the $△ABC$ and $BC$. Then $\theta$ is equal to

1. ${\tan }^{-1}\left(\frac{\cos B+\cos C-1}{\sin C-\sin B}\right)$
2. ${\tan }^{-1}\left(\frac{\sin C-\sin B}{\cos B+\cos C-1}\right)$
3. ${\tan }^{-1}\left(\frac{r-R\cos A}{r\cot \frac{B}{2}-R\sin A}\right)$
4. ${\tan }^{-1}\left(\frac{r\cos A-R}{r\sin A-R\cot \frac{B}{2}}\right)$

Q. In a $△ABC$, if $\cot A\cot B\cot C>0$. then the $△$ is

1. Acute angled
2. Right angled
3. Abtuse angles
4. Does not exist.

Q. Let $r$ and $R$ be the inradius and the circumradius of a $△ABC$. Let $\theta$ be the angle between the line joining the incentre and the circumcentre of the $△ABC$ and $BC$. Then $\theta$ is equal to

1. ${\tan }^{-1}\left(\frac{\cos B+\cos C-1}{\sin C-\sin B}\right)$
2. ${\tan }^{-1}\left(\frac{\sin C-\sin B}{\cos B+\cos C-1}\right)$
3. ${\tan }^{-1}\left(\frac{r-R\cos A}{r\cot \frac{B}{2}-R\sin A}\right)$
4. ${\tan }^{-1}\left(\frac{r\cos A-R}{r\sin A-R\cot \frac{B}{2}}\right)$

Q. If in an isosceles right angled triangle the area of circumcircle is 16 times the area of incircle, also all points circumcenter, orthocenter, incenter, centroid lie in the 1st quadrant, orthocenter being at origin. Find the distance between circumcenter and incenter.

1. 4R - 2 $\sqrt{2}$ r
2. 4r - 2 $\sqrt{2}$ r
3. 4R - 2 $\sqrt{2}$ R
4. 4 - 2 $\sqrt{2}$ r

Q. STATEMENT 1: lf two angles of a triangle satisfy the equation ${\tan }^2\theta +p\tan \theta -1=0$, then the triangle is obtuse angled for all values $p$.
STATEMENT 2: In a $\Delta ABC,\sum \tan A\tan B=1$

1. Statement-1 is true, Statement-2 is true, Statement-2 is correct explanation of Statement- 1
2. Statement-1 is true, Statement-2 is false
3. Statement-1 is false, Statement-2 is true
4. Statement-1 is true, Statement-2 is true, Statement-2 is not correct explanation for Statement-1

Q. If area of the triangle formed by the points $(1,-2),(3,4)$ and $(2,0)$ is half of the area of triangle formed by the points $(3,4),(2,0)$ and $(\alpha ,{\alpha }^2)$, then distance between centroids of respective triangles is

1. $\sqrt{\frac{35}{3}}$ units
2. $2$ units
3. $\sqrt{\frac{37}{3}}$ units
4. $\sqrt{\frac{39}{3}}$ units

Q. If ABC is an acute-angled triangle with circumcenter O and Orthocenter H. If AO = AH, then angle A = .

1. $\frac{\pi }{6}$
2. $\frac{\pi }{8}$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{4}$

Q. In a right angled triangle PQR, Circumcenter and orthocenter are at a distance of ------ units from each other. (Where R is the circumradius)

1. 4R

2. R

3. 2R

4. 3R

Q. Let $A(1,0),B(6,2)$, $C(\frac{3}{2},6)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle $ABC$ such that the triangles $APC$, $APB$ and $BPC$ have equal areas, then the length of the line segment $PQ$, where $Q$ is the point $(-\frac{7}{6},-\frac{1}{3})$, is

Q. Let $A(1,0),B(6,2)$, $C(\frac{3}{2},6)$ be the vertices of a triangle $ABC$. If $P$ is a point inside the triangle $ABC$ such that the triangles $APC$, $APB$ and $BPC$ have equal areas, then the length of the line segment $PQ$, where $Q$ is the point $(-\frac{7}{6},-\frac{1}{3})$, is

Q. In a right angled triangle PQR, Circumcenter and orthocenter are at a distance of ------ units from each other. (Where R is the circumradius)

1. 3R

2. R

3. 2R

4. 4R

Q. The corresponding interplanar spacing is ($inpm$)?

Q.

In an equilateral triangle sum of the distances of orthocenter from all the vertices is equal to -
(Where 'R' is the circumradius of triangle)

1. R
   

2. 2R

3. 3R

4. 4R