3 Marks Question

Question 13 Marks

The sides of a rectangle are in the ratio $4 : 5$ and its perimeter is $180\ cm$. Find its sides.

Answer

Let the length be $4x\ cm$ and the breadth be $5x\ cm$.
Perimeter of the rectangle $=180\ cm.$
Perimeter of the rectangle $= 2(l + b) 2(l + b)$
$2(l + b) = 180$
$\Rightarrow 2(4x + 5x) = 180$
$\Rightarrow 2(9x) = 180$
$\Rightarrow 18x = 180$
$\Rightarrow x = 10$
$\therefore$ Length $= 4x\ cm = 4 \times 10 = 40\ cm$.
Breadth $= 5x\ cm = 5 $× $10 = 50\ cm$.

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Question 23 Marks

The diagonals of a rhombus are $16\ cm$ and $12\ cm$. Find the length of each side of the rhombus.

Answer

All the sides of a rhombus are equal in length. The diagonals of a rhombus intersect at $90^\circ $.
The diagonal and the side of a rhombus form right triangles.

In $\triangle\text{APB}:$
$AB^2= AO^2+ OB^2$
$= 82 + 62$
$ = 64 + 36 $
$= 100 $
$AB = 10\ cm,$
Therefore, the length of each side of the rhombus is $10\ cm$.

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Question 33 Marks

The angles of a quadrilateral are in the ratio $1 : 2 : 3 : 4$. Find the measure of each angle.

Answer

Let the angles be $(x)^\circ , (2x)^\circ , (3x)^\circ $ and $(4x)^\circ . (x)^\circ , (2x)^\circ , (3x)^\circ$ and $(4x)^\circ $.
Sum of the angles of a quadrilateral is $360^\circ $.
$x + 2x + 3x + 4x = 360 10x = 360$
$\text{x}=\frac{360}{10}$$ x = 36 (2x)^\circ $
$= (2 \times 36)^\circ = 72^\circ (3x)^\circ $
$= (3 \times 36)^\circ = 108^\circ (4x)^\circ $
$ = (4 \times 36)^\circ = 144^\circ $
The angles of the quadrilateral are $36^\circ , 72^\circ , 108^\circ $ and $144^\circ . 36^\circ , 72^\circ , 108^\circ $ and $144^\circ $.

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Question 43 Marks

Two adjacent angles of a parallelogram are in the ratio $2 : 3$. Find the measure of each of its angles.

Answer

Let the two adjacent angles of the parallelogram be $(2x)^\circ$ and $(3x)^\circ$ .
Sum of any two adjacent angles of a parallelogram is $180^\circ$ .
$\therefore$ $2x + 3x = 180$
$\Rightarrow 5x = 180$
$\Rightarrow x = 36 (2x)^\circ $
$= (2 \times 36)^\circ = 72^\circ (3x)^\circ$
$= (3 \times 36)^\circ = 108^\circ$
Measures of the angles are $72^\circ$ and $108^\circ$ .

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