Rectangle

Q. If the midpoints of the sides of a quadrilateral are joined in an order (in succession), prove that the resulting quadrilateral is a parallelogram.

Q. ABCD is a rectangle as shown in the figure. Then AC = ___ $\times$ OB.

Q. In a quadrilateral ABCD, If bisectors of the $\angle ABC$ and $\angle ADC$ meet on the diagonal AC, prove that the bisectors of $\angle BAD$ and $\angle BCD$ will meet on the diagonal BD.

Q. Q: In a quadrilateral ABCD, angle A + angle C = 2*(angle B + angle D). if angle A = 140 degree and angle D = 60 degree, then angle B is?

Q. Prove that a parallelogram which is not a rectangle is not cyclic.

Q. The bisectors of opposite angles are parallel to each other.

Q. In the given figure, $AB\parallel CE$ and $DF\parallel CB$. Find the value of $x$ and $y$

Q. $ABCD$ is a rectangular with $AD=12\text{cm}$ and $DC=20\text{cm}.$ Line segment $DE$ is drawn making an angle of ${30}^{\circ }$ with $AD,$ intersecting $AB$ in $E.$ Find the length of $DE$ and $AE.$

Q. Assertion :If a, b, c $ϵR$ and $a\ne b\ne c$ and x, y, z are non zero. Then the system of equations $ax+by+cz=0$
$bx+cy+az=0$
$cx+ay+bz=0$ has infinite solutions. Reason: If the homogeneous system of equations has non trivial solution, then it has infinitely many solutions.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Assertion is incorrect but Reason is correct

Q. In figure given below, D and E are mid-points of the sides $BC$
and CA respectively of a
$△ABC$, right angled at $C$.
Prove that:
(i) $4{AD}^2=4{AC}^2+{BC}^2$
(ii) $4{BE}^2=4{BC}^2+{AC}^2$
(iii) $4({AD}^2+{BE}^2)=5{AB}^2$

Q. A kite in the shape of a square with a diagonal $32cm$ and an isosceles triangle of base $8cm$ and sides $6cm$ each is to be made of three different shades as shown in Fig. How much paper of each shade has been used in it?