If the Diagonals of a Quadrilateral Bisect Each Other then it is a Parallelogram.

Q.

From the given figure, what can be said about quadrilateral ABCD?

1. Rhombus

2. Parallelogram

3. Kite

4. Square

Q. Show that if the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.

Q. In an acute-angled triangle ABC, the altitudes from A, B and C when extended intersect the circumcircle again at points $A_1,B_1,C_1$ , respectively. If $\angle ABC={45}^o$ then $\angle A_1,B_1,C_1$ :

1. ${45}^o$
2. ${60}^o$
3. ${90}^o$
4. ${135}^o$

Q. In the figure, $\angle PQR={100}^o$, where $P,Q$ and $R$ are points on a circle with centre $O$. Find $\angle OPR$.

Q. If the diagonal of a quadrilateral bisects the opposite angles as shown, then the given quadrilateral is a _______.

1. Kite
2. Rectangle
3. Square
4. Rhombus

Q. In the given figure, ACDH is a parallelogram, EFGH is a square and the straight line BEG is parallel to CD. Find $\angle CDE$

1. ${45}^0$
2. ${125}^0$
3. ${135}^0$
4. ${185}^0$

Q. A triangle and a parallelogram are on the same base and between the same parallel lines they have:

1. Equal area
2. None of these
3. Equal perimeter
4. Both

Q. The perimeter of triangle ABC is 38 cm. AB=AC and angle(AB) greater than angle (BC) by 7 cm. Find the length of the sides of triangle ABC.

Q. If the diagonals of a quadrilateral bisect one another at right angles, then the quadrilateral is a:

1. rhombus
2. trapezium
3. rectangle
4. parallelogram

Q. The figure above shows a parallelogram $ABCD$ with $AC=3$ and $AD=5$. Calculate the area of parallelogram $ABCD$.

1. $12$
2. $18$
3. $15$
4. $20$

Q.

Assertion : In $\Delta ABC$, median AD is produced to X such that AD = DX. Then, ABXC is a parallelogram.
Reason : Diagonals AX and BC bisect each other at right angles.
Which of the following is correct?

1. If both assertion and reason are true and reason is the correct explanation of assertion
2. If assertion is false but reason is true
3. If both assertion and reason are true but reason is not the correct explanation of assertion
4. If assertion is true but reason is false

Q. In the adjoining figure, $O$ is the centre of a circle. If $AB$ and $AC$ are chords of the circle such that $AB=AC,OP\perp AB$ and $OQ\perp AC$, prove that $PB=QC$

Q. In the figure, $AC$ is the diameter of the circle with centre $O$. Chord $BD\perp AC$ , Write down angles $p,q$ and $r$ in terms of $x$. $\angle ACB=q$