Quadrilaterals Test

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Quadrilaterals Test - 26

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Weekly Quiz Competition

Question 1
1 / -0

Complete the following statement .
Number of measurements required to construct a rectangle are_______.

A
4

B
3

C
2

D
1

Solution

If we know the length and breadth of the rectangle, we can construct it since a rectangle has opposite sides equal in length and each of the four angles is equal to $$ {90}^{0} $$
Hence, number of measurements required to construct a rectangle is $$2$$.
Question 2
1 / -0

Which of the following statements is true?

A
The diagonals of a rectangle are perpendicular

B
The diagonals of a rhombus are equal

C
Every square is a rhombus

D
None of these

Solution

Every square is a rhombus is the correct statement.
A rhombus is a quadrilateral with all sides equal is length. A square is a quadrilateral with all sides equal in length and all interior angles right angle.
Thus a rhombus is not a square when the angles are all right angles. a square is a rhombus since all four of its sides are of the same length.
Hence, the answer is every square is a rhombus.
Question 3
1 / -0

Consider the following statements and state which one is true and which one is false:
(1) The bisectors of all the four angles of a parallelogram enclose a rectangle.
(2) The figure formed by joining the midpoints of the adjacent sides of the rectangle is a rhombus.
(3) The figure formed by joining the midpoints of the adjacents sides of a rhombus is square.

A
1 and 2

B
2 and 3

C
3 and 1

D
1,2 and 3

Solution

1) The bisectors of all the four angles of a parallelogram enclose a rectangle. The true statement.
2) The figure formed by joining the midpoints of the adjacent sides of the rectangle is a rhombus. The true statement.
3) The figure formed by joining the midpoints of the adjacent sides of a rhombus is square. It would be a rectangle. It is not true.
Then 1 and 2 are true
Question 4
1 / -0

What is the value of $$x$$ in the given figure?

A
$$b-a-c$$

B
$$b-a+c$$

C
$$b+a-c$$

D
$$(a+b+c)$$

Solution

Reflex $$\angle BCD={360}^{o}-b$$
In quad $$ABCD$$
$$\angle A+\angle B+reflex \angle b+\angle D={360}^{o}$$
$$\Rightarrow$$ $$a+c+{360}^{o}-b+x={360}^{o}$$
$$\Rightarrow$$ $$x={360}^{o}-{360}^{o}-a-c+b$$
$$=b-a-c$$
Question 5
1 / -0

A quadrilateral in which only one pair of opposite sides are parallel is called a ..........

A
square

B
rectangle

C
rhombus

D
trapezium

Solution

$$\Rightarrow$$ We know that in square, rectangle and rhombus two pairs of opposite sides are parallel, so answer can not be square, rectangle or rhombus.
$$\Rightarrow$$ We also know the properties of trapezium in which only one pair of opposite sides are equal.
$$\Rightarrow$$ So, correct answer is option D trapezium.
Question 6
1 / -0

Which of the following statement(s) is/are true?

A
A parallelogram in which two adjacent angles are equal is a rectangle

B
A quadrilateral in which both pairs of opposite angles are equal is parallelogram

C
In a parallelogram the number of acute angles is zero (or) two

D
All the above

Solution

$$i).$$ A parallelogram in which two adjacent angles are equal is a rectangle.
i.e, a rectangle is a parallelogram in which one angle is $$90°.$$
$$ii).$$ A quadrilateral in which both pairs of opposite angles are equal is parallelogram.
$$iii).$$ In a parallelogram the number of acute angles is zero or two and the no. of obtuse angles are two.
The sum of adjacent angles will be $$180°.$$
$$\therefore$$ All the options are correct.
Hence, the answer is all the above.
Question 7
1 / -0

In a parallelogram, the sum of adjacent angles is:

A
$$90^{\circ}$$

B
$$180^{\circ}$$

C
$$270^{\circ}$$

D
$$360^{\circ}$$

Solution

$$ABCD$$ is a parallelogram.
$$\therefore$$ $$AB$$ $$\parallel$$ $$DC$$ and $$AD$$ $$\parallel BC$$
$$\therefore$$ $$\angle$$$$d$$ $$+$$ $$\angle$$ $$e$$ (corresponding angle)
$$\angle$$$$a$$ $$+$$ $$\angle e$$ = $$180$$$$^\circ$$ (supplementary angle or linear pair)
$$\angle$$$$a $$ $$+$$ $$\angle$$$$d$$ = 180$$^\circ$$ ($$\because$$ $$\angle$$$$d$$ = $$\angle$$$$e$$)
$$\therefore$$ sum of adjacent angle are $$180$$$$^\circ$$
Ans : B)$$ 180$$$$^\circ$$
Question 8
1 / -0

In the diagram, $$ABD$$ and $$BCD$$ are isosceles triangles triangles, where $$AB=BC=BD$$. The special name that is given to quadrilateral $$ABCD$$ is:

A
rectangle

B
kite

C
parallelogram

D
trapezium

Solution

In $$\Delta BAD$$,
$$\angle BDA=\angle BAD=57^{\circ}$$ (isosceles triangle property)
In $$\Delta BDC$$,
$$\angle BCD=\angle BDC=66^{\circ}$$ (isosceles triangle property)
$$\therefore \displaystyle \angle D=\angle ADB + \angle BDC = 57^{\circ}+66^{\circ}=123^{\circ}$$ and
$$\angle A+\angle D=57^{\circ}+123^{\circ}=180^{\circ}$$
Also, $$\angle D+ \angle C=123^{\circ}+66^{\circ}=189^{\circ}$$
Hence, by the property that corresponding interior angles are supplementry, lines are parallel, we have $$AB \parallel DC$$ and $$AD \not \parallel BC$$
Hence, $$ABCD$$ is a trapezium.
Question 9
1 / -0

If the bisectors of the angles $$A, B, C$$ and $$D$$ of a quadrilateral meet at $$O$$, then $$\angle AOB$$ is equal to:

A
$$\angle C+\angle D$$

B
$$\displaystyle \frac{1}{2}(\angle C+\angle D)$$

C
$$\displaystyle \frac{1}{2}\angle C+\frac{1}{3}\angle D$$

D
$$\displaystyle \frac{1}{3}\angle C+\frac{1}{2}\angle D$$

Solution

In quad. $$ABCD$$

$$\angle A+\angle B+\angle C+\angle D=360^{\circ}$$

$$\Rightarrow \angle A+\angle B=360^{\circ}-(\angle C+\angle D)$$ ...(i)

In $$\Delta AOB$$, we know

$$\angle AOB+\angle OAB+\angle OBA=180^{\circ}$$

$$\angle AOB=180^{\circ}-(\angle OAB+\angle OBA)$$

$$=\displaystyle 180^{\circ}-\frac{1}{2}(\angle A+\angle B)$$

($$\because $$ $$OA$$ bisects $$\angle A$$ and $$OB$$ bisects $$\angle B$$)

Thus from (i),
$$\angle AOB=180^{\circ}-\dfrac{1}{2}\left \{ 360^{\circ}-(\angle C+\angle D) \right \}$$

$$=\displaystyle 180^{\circ}-180^{\circ}+\dfrac{1}{2}(\angle C+\angle D)$$

$$=\displaystyle \frac{1}{2}(\angle C+\angle D)$$
Question 10
1 / -0

In $$\triangle PQR, QR=10, RP=11$$ and $$PQ=12.$$ $$D$$ is the midpoint of $$PR$$. $$DE$$ is drawn parallel to $$PQ$$ meeting $$QR$$ in $$E.$$ $$EF$$ is drawn parallel to $$RP$$ meeting $$PQ$$ in $$F.$$ What is the length of $$DF\: ?$$

A
$$\displaystyle \frac{11}{2}$$

B
$$6$$

C
$$\displaystyle \frac{33}{4}$$

D
$$5$$

Solution

Given,
$$D$$ is the midpoint of $$PR$$.
and $$DE \parallel PQ$$.
$$ \implies E$$ is the midpoint of $$QR$$.

We know,
Midpoint Theorem states that the segment joining two sides of a triangle at the midpoints of those sides is parallel to the third side and is half the length of the third side.

Similarly, $$F$$ is the midpoint of $$PQ$$
Then, from Midpoint Theorem,
$$DF || AR$$ and $$DF=\dfrac{1}{2}QR$$

Since, $$QR=10$$.
$$\implies$$ $$DF=\dfrac{1}{2}(10)=5$$.

Hence, option $$D$$ is correct.

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Quadrilaterals Test - 26

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Quadrilaterals Test - 26

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Question 1

4

3

2

1

Solution

Question 2

The diagonals of a rectangle are perpendicular

The diagonals of a rhombus are equal

Every square is a rhombus

None of these

Solution

Question 3

1 and 2

2 and 3

3 and 1

1,2 and 3

Solution

Question 4

$$b-a-c$$

$$b-a+c$$

$$b+a-c$$

$$(a+b+c)$$

Solution

Question 5

square

rectangle

rhombus

trapezium

Solution

Question 6

A parallelogram in which two adjacent angles are equal is a rectangle

A quadrilateral in which both pairs of opposite angles are equal is parallelogram

In a parallelogram the number of acute angles is zero (or) two

All the above

Solution

Question 7

$$90^{\circ}$$

$$180^{\circ}$$

$$270^{\circ}$$

$$360^{\circ}$$

Solution

Question 8

rectangle

kite

parallelogram

trapezium

Solution

Question 9

$$\angle C+\angle D$$

$$\displaystyle \frac{1}{2}(\angle C+\angle D)$$

$$\displaystyle \frac{1}{2}\angle C+\frac{1}{3}\angle D$$

$$\displaystyle \frac{1}{3}\angle C+\frac{1}{2}\angle D$$

Solution

Question 10

$$\displaystyle \frac{11}{2}$$

$$6$$

$$\displaystyle \frac{33}{4}$$

$$5$$

Solution

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