3 Marks Question

Question 13 Marks

Diagonals $A C$ and $B D$ of a parallelogram $A B C D$ intersect each other at $O$. If $O A=3 cm$ and $O D=2 cm$, determine the lengths of $A C$ and $B D$.

Answer

Given, $ABCD$ is a paralleelogram $OA = 3cm$ and $OD = 2cm$

We now that, diagonals of a parallelogram bisect each other.
$\therefore$ Diagonal $AC = 2 OA = 6cm$ [$\because AO = OC$] and diagonal $BD = 2 OD = 4cm$ [$\because BO = OD$] Hence, the length of the diagonais $AC$ and $BD$ are $6cm$ and $4cm$, repectively

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Question 23 Marks

$ABCD$ is a trapezium in which $AB \| DC$ and $\angle\text{A}=\angle\text{B}=45^\circ.$ Find angles $C$ and $D$ of the trapezium.

Answer

Given, $A B C D$ is a trapezium and whose parallel sides in the figure are $A B$ and $D C$. Since, $A B \| C D$ and $B C$ is transversal, then sum of two cointerior angles is $180^{\circ}$
.

$\therefore\ \angle\text{B}+\angle\text{C}=180^\circ$
$\Rightarrow\ \angle\text{C}=180^\circ-\angle\text{B}=180^\circ-45^\circ$ [$\because\angle\text{B}=45^\circ$ given] $\Rightarrow\ \angle\text{C}=135^\circ$ Similarly, $\angle\text{A}+\angle\text{D}=180^\circ$ [sum of cointerior is 180^\circ ] $\Rightarrow\ \angle\text{D}=180^\circ-45^\circ$ [$\because\angle\text{A}=45^\circ$ given] $\Rightarrow\angle\text{D}=135^\circ$ Hence, angles $C$ and $D$ are $135^\circ$ each

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Question 33 Marks

All the angles of a quadrilateral are equal. What special name is given to this quadrilateral?

Answer

We know that, sum of all angles in a quadrilateral is $360^\circ $. If $ABCD$ is a quadrilateral, $\angle\text{A}+\angle\text{B}+\angle\text{C}+\angle\text{D}=360^\circ...(\text{i})$ But it is given all angles are equal. $\angle\text{A}=\angle\text{B}=\angle\text{C}=\angle\text{D}$ from eq$. ...(i)$
$\angle\text{A}+\angle\text{A}+\angle\text{A}+\angle\text{A}=360^\circ$
$\Rightarrow\ 4\angle\text{A}=360^\circ$
$\angle\text{A}=90^\circ$ So, all angles of a quadrilateral are $90^\circ $.
Hence, given quadrilateral is a rectangle.

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