Intercept Theorem

Q.

If $a,b,c$ are position vectors of three-collinear points such that $xa+yb+zc=0$and atleast one scalar $x,y,z,\ne 0,$ then

Q.

$O$ is the orthocentre of $△ABC$. If $\angle BAC={70}^{\circ },$ then $\angle BOC=$

1. ${100}^{\circ }$

2. ${120}^{\circ }$

3. ${110}^{\circ }$

4. ${140}^{\circ }$

Q.

In quadrilateral ABCD, its diagonals AC and BD intersect at point O such that $\frac{OC}{OA}=\frac{OD}{OB}=\frac{1}{3}$

If CD =4.5 cm; find the length of AB.

1. 12 cm

2. 14.5 cm

3. 13.5 cm

4. 11.8 cm

Q.

A circular disc of radius 6 cm is divided into three sectors with central angles 120^o, 150^o and 90^o. The ratio of the arc of the three sectors is

1. 4:5:3

2. 3:4:5

3. 6:5:4

4. 4:5:6

Q.

In a parallelogram ABCD, the line drawn through a point P on AB; parallel to BC, meets AC at Q. The line through Q, parallel to AB meets AD at R. Then, $\frac{AP}{PB}=$

1. $\frac{AQ}{QC}$

2. $\frac{AR}{RD}$

3. $\frac{RQ}{QP}$

4. $\frac{AC}{AP}$

Q. The number of arcs that have to be drawn on ray BA so as to divide BC in the ratio of 2:3.

1. 5

Q. In a parallelogram ABCD, the line drawn through a point P on AB; parallel to BC, meets AC at Q. The line through Q, parallel to AB meets AD at R. Then, prove that $\frac{AP}{PB}=$ $\frac{AQ}{QC}=$ $\frac{AR}{RD}$.

Q.

In quadrilateral ABCD, its diagonals AC and BD intersect at point O such that $\frac{OC}{OA}=\frac{OD}{OB}=\frac{1}{3}$

If CD =4.5 cm; find the length of AB.

1. 12 cm

2. 14.5 cm

3. 13.5 cm

4. 11.8 cm

Q. To divide a line segment, the constructed ray must be at an obtuse angle to the line segment.

1. True
2. False

Q. We only need to check if the corresponding angles are equal for two triangles to be similar.

1. True
2. False

Q.

AB and CD are parallel lines.

Then,

1. $AP\times PC=BP\times BD$

2. $AP\times PC$

   $=(BP{)}^2\times PD$

3. $AP\times PC$

   $=(BP{)}^2\times (PD{)}^2$

4. $AP\times PC$

   $=\frac{BP}{PD}$

Q.

$\text{ABCD is a parallelogram with diagonal}\text{AC. If a line XY is drawn such that it cuts}\text{the side AD at point Z and XY || AB, then}\frac{BX}{XC}\text{can be equal to:}$

1. $\frac{AY}{AC}$

2. $\frac{DZ}{AZ}$

3. $\frac{AZ}{ZD}$

4. $\frac{AY}{YC}$