3 Marks Question

Question 13 Marks

Diagonals AC and BD of a parallelogram ABCD intersect each other at O. If OA = 3cm and OD = 2cm, determine the lengths of AC and BD.

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, ABCD is a trapezium and whose parallel sides in the figure are AB and DC. Since, AB || CD and BC is transversal, then sum of two cointerior angles is 180°.

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

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