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Quadrilaterals

Class 8th Mathematics (old) MHB Solution
Exercise 14
  1. Recall the properties of quadrilaterals and fill in the blanks.
  2. Recall the properties of quadrilaterals and fill in the blanks.
  3. Recall the properties of quadrilaterals and fill in the blanks.
  4. Recall the properties of quadrilaterals and fill in the blanks.
  5. Recall the properties of quadrilaterals and fill in the blanks.
  6. Recall the properties of quadrilaterals and fill in the blanks.
  7. Recall the properties of quadrilaterals and fill in the blanks.
  8. Recall the properties of quadrilaterals and fill in the blanks.
Exercise 15
  1. Find the length of diagonal QS of a square PQRS if the length of diagonal PR is…
  2. If, in the square ABCD, l(AB) = 4.5 cm, what are the lengths of the other sides…
  3. The diagonals seg DF and seg EG of the square DEFG intersect each other in point…
  4. The segments XZ and YW are diagonals of the square XYZW. M is their point of…
  5. In the square HDFC, if l(HF) = 5 cm, find l(CD).
Exercise 16
  1. In rectangle PQRS, l(PQ) = 7 cm, l(PS) = 9 cm. Find l(QR) and l(SR).…
  2. The diagonals AC and BD of rectangle ABCD intersect in point O. If m∠CAB = 25°…
  3. The diagonals AC and BD of rectangle ABCD intersect in point K. If l(AK) = 3.5…
  4. The diagonals XZ and WY of rectangle XYZW intersect in point M. If l(XZ) = 8cm…
  5. In quadrilateral LMNP, m∠L = m∠M = m ∠N = 90°. Find m ∠ P. What type of…
Exercise 17
  1. If the length of one side of a rhombus is 7.5 cm, find the lengths of the…
  2. The diagonals seg XZ and seg YW of the rhombus XYZW intersect at point P. If…
  3. If m∠QPS in the rhombus PQRS is 65°, find m∠QRS.
  4. The diagonals AC and BD of a rhombus ABCD intersect at point O. Find m∠AOD and…
  5. In the rhombus KING, m∠K = 70° and m∠I = 110°. Find the measures of the other…
Exercise 18
  1. The diagonals LN and MT of parallelogram LMNT intersect in point O. If l(MO) =…
  2. In parallelogram PQRS, m∠Q = 130°. Find the measures of the other angles of…
  3. The measures of the opposite angles of a parallelogram are (3x - 2)° and (50 -…
Exercise 14
Question 1.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1) Congruent


Explanation: Opposite sides of parallelogram are equal and parallel


(2) congruent


Explanation: Opposite angles of parallelogram are equal


(3) diagonals


Explanation: Diagonals of parallelogram bisect each other
Question 2.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1) bisect each other at right angles.


Explanation: Both diagonals of rhombus bisect each other perpendicularly


(2) perpendicular bisector


Explanation: Act as perpendicular bisector as both equally divides other in 2 parts by making angle of 90


(3) opposite


Explanation: In rhombus opposite angles are congruent and consecutive angles are supplementary
Question 3.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1) opposite sides


Explanation: The opposite sides are equal and parallel in rectangle


(2) congruent


Explanation: The diagonals of rectangle are equal and bisect each other


(3) diagonal


Explanation: Diagonals bisect each other of rectangle
Question 4.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1) congruent


Explanation: Diagonals are equal and perpendicular bisector of square


(2) diagonals


Explanation: Diagonals of square bisect each other


(3) perpendicular bisector


Explanation: In square both diagonal divide each other perpendicularly
Question 5.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1)


Explanation: Because in kite 2 pair of sides have equal length


(2)


Explanation: Because in kite 2 pair of sides have equal length


(3) 90


Explanation: Diagonals of kite cross at 90 but both don’t bisect each other


(4) kite


Explanation: It is kite having only one diagonal bisected by other and having 2 pairs of sides are equal
Question 6.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1) 90


Explanation: Due to the perpendicular bisector of diagonals of rhombus


(2)


Explanation: Because diagonal bisect each other in rhombus


(3)


Explanation: Because of opposite angles of rhombus are equal
Question 7.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1) side DF


Explanation: Because of opposite sides of rectangle are equal


(2)


Explanation: Because of diagonals of rectangle are equal


(3)


Explanation: Because of diagonals of rectangle bisect each other
Question 8.

Recall the properties of quadrilaterals and fill in the blanks.
Answer:

(1)


Explanation: Because of diagonals of square are equal


(2) 90


Explanation: Because of diagonals of square are perpendicular bisector of each other


(3)
Explanation: Because of diagonals of square are equal and bisectors of each other
Exercise 15
Question 1.

Find the length of diagonal QS of a square PQRS if the length of diagonal PR is 8 cm.
Answer:

The figure of square PQRS is given below:

As we know, In a square,


All the sides are equal.


i.e. PS = SR = RQ = PQ



Also, both the diagonals in a square are equal.


i.e. PR = QS


Now, PR = 8 units.


⇒ QS = 8 units.
Question 2.

If, in the square ABCD, l(AB) = 4.5 cm, what are the lengths of the other sides of the square?
Answer:

The square ABCD is shown below:



As we know, Length of all sides of square are equal


If AB = 4.5cm


Then AB = BC = CD = DA


⇒ BC = 4.5cm


⇒ CD = 4.5cm


⇒ AD = 4.5cm
Question 3.

The diagonals seg DF and seg EG of the square DEFG intersect each other in point M. If l(DM) = 7 cm, find l(EG).
Answer:

The figure is given below:



As we know that diagonals of square are equal and bisect each other


i.e. Diagonal DF = Diagonal EG [equal diagonals]


AND DM = MF [diagonals bisect]


DF = DM+MF


DF = 2× DM = 2× 7cm ×


DF = 14cm


If Diagonal DF = EG


EG = 14cm
Question 4.

The segments XZ and YW are diagonals of the square XYZW. M is their point of intersection. Find m∠XMY.
Answer:

The figure is shown below:


As we all know that diagonals of square bisect each other perpendicularly ,


Hence the angle ∠XMY made by the intersection of the diagonals XZ and WY will be 90
Question 5.

In the square HDFC, if l(HF) = 5 cm, find l(CD).
Answer:

The figure is given below:


As we know that the diagonals of square are equal


∵ HF = CD [by above statement]


⇒ CD = HF = 5cm
Exercise 16
Question 1.

In rectangle PQRS, l(PQ) = 7 cm, l(PS) = 9 cm. Find l(QR) and l(SR).
Answer:

The figure is given below:



As we know that,


In rectangle,


Length of opposite sides are equal.


Then PQ = SR [by above statement]


And PS = QR [by above statement]


Hence:


SR = 7cm


QR = 9cm
Question 2.

The diagonals AC and BD of rectangle ABCD intersect in point O. If m∠CAB = 25° find m∠DAC and m∠ACD.
Answer:


As we know that, In rectangle every angle is 90


Hence,


∠CAB+∠DAC = 90°


⇒ ∠DAC = 90° - ∠CAB


⇒ ∠DAC = 90°-25°


⇒ ∠DAC = 65°


As we know that, In rectangle opposite sides are parallel


∠CAB = ∠ACD [By alternate interior angles]


Hence;
Question 3.

The diagonals AC and BD of rectangle ABCD intersect in point K. If l(AK) = 3.5 cm, then l(KC) = ? and l(AC) = ?
Answer:


As we know that, the diagonal of rectangle bisect each other


AK = KC [by above statement]


KC = 3.5cm


AND, AC = KC+AK


AC = 3.5cm+3.5cm


AC = 7cm
Question 4.

The diagonals XZ and WY of rectangle XYZW intersect in point M. If l(XZ) = 8cm then l(XM) find and l(YM).
Answer:


As we know that, the diagonal of rectangle are equal and bisect each other


Then XM = ZM [by above statement]


Diagonal XZ = XM+ZM


XZ = 2× XM


IF XZ = 8cm


Then



XM = 4cm
Question 5.

In quadrilateral LMNP, m∠L = m∠M = m ∠N = 90°. Find m ∠ P. What type of quadrilateral is □LMNP?
Answer:

As we know that, sum of all angles of quadrilateral is 360


Then;


[by above statement]





If all angles of quadrilateral is 90 then


It may be a RECTANGLE OR A SQUARE
Exercise 17
Question 1.

If the length of one side of a rhombus is 7.5 cm, find the lengths of the remaining sides.
Answer:

As we know, In a rhombus


All the sides are equal.


if one side is 7.5cm then all other sides will be 7.5cm
Question 2.

The diagonals seg XZ and seg YW of the rhombus XYZW intersect at point P. If l(XP) = 8 cm, find l(XZ).
Answer:


The diagonals of rhombus are perpendicular bisectors of each other


XP = PZ [by above statement]


XZ = XP+PZ [by diagram]


XZ = 2× XP


XZ = 2× 8cm = 16cm
Question 3.

If m∠QPS in the rhombus PQRS is 65°, find m∠QRS.
Answer:


In rhombus opposite angles are congruent, and consecutive angles are supplementary


[by above statement]


l
Question 4.

The diagonals AC and BD of a rhombus ABCD intersect at point O. Find m∠AOD and m∠BOC.
Answer:


The diagonals of rhombus are perpendicular bisectors of each other



[by vertically opposite angle


as in the diagram ]
Question 5.

In the rhombus KING, m∠K = 70° and m∠I = 110°. Find the measures of the other angles of the rhombus KING.
Answer:


In rhombus opposite angles are congruent, and consecutive angles are supplementary


[defined opposite angles as in above Statement ]
Exercise 18
Question 1.

The diagonals LN and MT of parallelogram LMNT intersect in point O. If l(MO) = 5cm, l(LN) = 6cm, find l(OT) and l(NO).
Answer:


As we know that, Diagonal of the parallelogram bisect each other


MO = OT


NO = LO [BY above statement]


Hence ;


OT = 5cm


And


If LN = LO+NO [by diagram]


Then, LN = 2NO = 6cm


NO =



Question 2.

In parallelogram PQRS, m∠Q = 130°. Find the measures of the other angles of □PQRS.
Answer:


As we all know that , Opposite angels are congruent Consecutive angles are supplementary in parallelogram.


[by above statement]


For opposite angles


= 130°


For consecutive angles


[by above statement]


[by above statement]


As


× 50°
Question 3.

The measures of the opposite angles of a parallelogram are (3x – 2)° and (50 – x)°. Find the measure of each angle of the parallelogram.
Answer:

As we all know that, Opposite angels are congruent Consecutive angles are supplementary in parallelogram.


For opposite angles


3x-2 = 50-x [by above statement]


3x+x = 50+2


4x = 52



Angles are 50-13 = 37°


And for Consecutive angles


Y+ 37° = 180°


Y = 143°


Hence angles are 37°,143°,37,143°