Tag: construction of parallelograms and rectangles

Questions Related to construction of parallelograms and rectangles

Two line segments, each $9\ cm$ long, bisect each other at right angles. Their end points are joined together. The shape formed is a:

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. Square

  2. Kite

  3. Trapezium

  4. rhombus

Show answer
Correct Option: A
Explanation:

Image result for Two line segments, each 9 cm long, bisect each other at right angles. Their end points are joined together. The shape formed is a:

Let $PS$ and $QR$ are the two line segments, each of $9$cm, and bisect each other at right angles.
By joining the end points of these line, we get a shape given in the figure.
In $\triangle POQ$, $\angle POQ=90^{o}$, $OP=OQ=4.5$
By using Pythagoras theorem,
$PQ^{2}=OP^{2}+OQ^{2}$
         $=(4.5)^2+(4.5)^2=40.5$
$\therefore\ PQ=6.36$
Similarly, $QS=6.36=RS=PS$
Thus, length of all sides is same and all angles are right angle.
Hence, the shape formed is square.

A square with side given can be constructed by using the property of its diagonals.

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

This statement is true 

We can use property that diagonals are at 45 degree with side and diagonals bisect each other at 90 degree.for construction of square.

Can we construct a rhombus $ABCD$ with $AB=4\ cm$? Its diagonal intersect at the point $O$ and $\angle OAB = 60^0$.

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. Yes

  2. No

  3. Sometimes yes

  4. Can't say

Show answer
Correct Option: A
Explanation:

Given : $AB=4$cm

Diagonal intersect at $O$ and $\angle OAB=60^{o}$ ....... $(1)$
Draw side $AB$ of $4$cm.
In a rhombus, all sides are equal and diagonals bisect the opposite angles
From $(1)$ we get, $\angle A=120^{o}$
$\implies \angle B=60^{o}$ ........... (Adjacent angles are supplementary)
Draw a side $AD$ from A of $4$cm such that $\angle BAD=120^{o}$
Now, from $D$, draw side $DC = 4$cm such that $\angle ADC=60^{o}$
And then join $B-C$ such that $BC=4$cm and $\angle DCB=120^{o}$.
At last we get a rhombus $ABCD$ with length of each side is $4$ cm and diagonals $AC$ and $BD$.
Hence, we can construct a rhombus with $AB=4\ cm$.

We cannot construct a square if:

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. a side is given

  2. a diagonal is given

  3. one angle is $90^0$

  4. None of these

Show answer
Correct Option: C
Explanation:

If a side is given then we can draw a square with the same side as given.

If diagonals are given, by joining the endpoints we can draw the square.
In square, all angles are of $90^{o}$.
If one angle is $90^{o}$ is given, we can't directly conclude that all the angles are $90^{o}$.

Hence, if one angle is $90^{o}$ then we cannot construct a square.

When given a square, the construction of an angle bisector at any vertex will create the diagonal of the square. 

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. True

  2. False

Show answer
Correct Option: A
Explanation:

This is statement is true

We know that diagonals of square bisects the angle. So angle bisector will be diagonal.

You are given the length of a diagonal of a rhombus and one of the angles of the rhombus. Which property of the rhombus will be used in the construction of this rhombus?

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. The lengths of the sides of a rhombus are equal.

  2. The angles of a rhombus are $90^\circ$

  3. Diagonal of a rhombus bisects the opposite angles.

  4. Diagonals of a rhombus are perpendicular bisectors of each other.

Show answer
Correct Option: C
Explanation:

$\Rightarrow$   We have given the length of diagonal of rhombus and one of angles of rhombus.

$\Rightarrow$  To construct an rhombus we will use the property that the diagonal of a rhombus bisect the opposite angle.
Because we know opposite angles of rhombus are equal, so it will be easier to construct rhombus.

If we have to construct a square $PQRS$ whose diagonal is $8 \sqrt 2$ cm then its side is equal to ?

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. $8$ cm

  2. $4\sqrt2$ cm

  3. $4$ cm

  4. $8\sqrt2$ cm

Show answer
Correct Option: A
Explanation:

If the diagonal of square is $a$, then its side $=\dfrac{a}{\sqrt2}$

If diagonal is $8\sqrt2 $ cm, then its side $=\dfrac{8\sqrt2}{\sqrt2}=8$ cm.

State the following statement is True or False
The side of a square is $\sqrt2$ times the diagonal of a square

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

The side of square is $\dfrac{1}{\sqrt2}$ times the diagonal of square.

State the following statement is True or False
We cannot construct the square if only diagonal is given

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. True

  2. False

Show answer
Correct Option: B
Explanation:

If $a$ is the diagonal of square then its side $=\dfrac{a}{\sqrt2}$

We can construct a square, with its side given.

Which of the following statements is true for a rhombus?

maths construction of polygons construction of parallelograms and rectangles construction of special quadrilaterals constructions related to a quadrilateral

  1. It has only two pair of equal sides.

  2. Two of its angles are at right angles.

  3. Its diagonals bisect each other at right angles.

  4. It is always a square.

Show answer
Correct Option: C
Explanation:

Rhombus is a flat shape with 4 equal straight sides.All sides have equal length.Opposite sides are parallel, and opposite angles are equal.The altitude is the distance at right angles to two sides.And the diagonals "p" and "q" of a rhombus bisect each other at right angles.
So (C) is correct.
Answer (C) Its diagonals bisect each other at right angles.