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Maths 4 Mark Question

Construction of Quadrilaterals · Maharashtra Board STD 8 (MSBSHSE - Semi English Medium). 21 questions.

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Question 14 Marks
Construct a parallelogram PQRS in which QR = 6cm, PQ = 4cm and $\angle\text{PQR}=60^\circ.$
Answer

Steps of construction:
Step 1: Draw PQ = 4cm.
Step 2: Make $\angle\text{PQR}=60^\circ.$
Step 2: With Q as the centre, draw an arc of 6cm and name that point as R.
Step 3: With R as the centre, draw an arc of 4cm and name that point as S.
Step 4: Join SR and PS.
Then, PQRS is the required parallelogram.
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Question 24 Marks
Construct a quadrilateral PQRS in which PQ = 5.4cm, QR = 4.6cm, RS = 4.3cm, SP = 3.5cm and diagonal PR = 4cm.
Answer

Steps of construction:
Step 1: Draw PQ = 5.4cm
Step 2: With P as the centre and radius equal to 4cm, draw an arc.
Step 3: With Q as the centre and radius equal to 4.6cm, draw another arc, cutting the previous arc at R.
Step 4: Join QR.
Step 5: With P as the centre and radius equal to 3.5cm, draw an arc.
Step 6: With R as the centre and radius equal to 4.3cm, draw another arc, cutting the previous arc at S.
Step 7: Join PS and RS.
Thus, PQRS is the required quadrilateral.
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Question 34 Marks
Construct a quadrilateral PQRS in which PQ = 6cm, QR = 5.6cm, RS = 2.7cm, $\angle\text{Q}=45^\circ$ and $\angle\text{R}=90^\circ.$
Answer

Steps of construction:
Step 1: Draw QR = 5.6cm.
Step 2: Make $\angle\text{Q}=45^\circ$ and $\angle\text{R}=90^\circ.$
Step 3: With Q as the centre, draw an arc of 6cm. Name that point as P.
Step 4: With R as the centre, draw an arc of 2.7cm. Name that point as S.
Step 6: Join P and S.
Then, PQRS is the required quadrilateral.
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Question 44 Marks
construct a quadrilateral ABCD in which AB = 3.4cm, CD = 3cm, DA = 5.7cm, AC = 8cm and BD = 4cm.
Answer

Steps of construction:
Step 1: Draw AB = 3.4cm.
Step 2: With B as the centre and radius equal to 4cm, draw an arc.
Step 3: With A as the centre and radius equal to 5.7cm, draw another arc, cutting the previous arc at D.
Step 4: Join BD and AD.
Step 5: With A as the centre and radius equal to 8cm, draw an arc.
Step 6: With D as the centre and radius equal to 3cm, draw another arc, cutting the previous arc at C.
Step 7: Join AC, CD and BC.
Thus, ABCD is the required quadrilateral.
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Question 54 Marks
Construct a parallelogram, one of whose sides is 4.4cm and whose diagonals are 5.6cm and 7cm. Measure the other side.
Answer
We know that the diagonals of a parallelogram bisect each other.
Steps of construction: Step 1: Draw AB = 4.4cm. Step 2: With A as the centre and radius 2.8cm, draw an arc. Step 3: With B as the centre and radius 3.5cm, draw another arc, cutting the previous arc at point O. Step 4: Join OA and OB. Step 5: Produce OA to C, such that OC = AO. Produce OB to D, such that OB = OD. Step 5: Join AD, BC, and CD. Thus, ABCD is the required parallelogram. The other side is 4.5cm in length.
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Question 64 Marks
Construct a rectangle ABCD whose adjacent sides are 11cm and 8.5cm.
Answer

Steps of construction:
Step 1: Draw AB = 11cm.
Step 2: Make
$\angle\text{A}=90^\circ$
$\angle\text{B}=90^\circ$
Step 3: Draw an arc of 8.5cm from point A and name that point as D.
Step 4: Draw an arc of 8.5cm from point B and name that point as C.
Step 5: Join C and D.
Thus, ABCD is the required rectangle.
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Question 74 Marks
Prove that the diagonals of a rhombus bisect each other at right angles.
Answer
Rhombus is a parallelogram.

Consider:
$\triangle\text{AOB}$ and $\triangle\text{COD}$
$\angle\text{OAB}=\angle\text{COD}$ (alternate angle)
$\angle\text{ODC}=\angle\text{OBA}$ (alternate angle)
$\angle\text{DOC}=\angle\text{AOB}$ (vertically opposite angles)
$\triangle\text{AOB}\cong\text{COB}$
$\therefore\text{AO}=\text{CO}$
$\text{OB}=\text{OD}$
Therefore, the diagonals bisects at O.
Now, let us prove that the diagonals intersect each other at right angles.
Consider $\triangle\text{COD}$ and $\triangle\text{COB}:$
CD = CB (all sides of a rhombus are equal)
CO = CO (common side)
OD = OB (point O bisects BD)
$\therefore\triangle\text{COD}\cong\triangle\text{COB}$
$\therefore\angle\text{COD}= \angle\text{COB}$ (corresponding parts of congruent triangles)
Further, $\angle\text{COD}+\angle\text{COB}=180^\circ$ (l inear pair)
$\therefore\angle\text{COD}=\angle\text{COB}=90^\circ$
It is proved that the diagonals of a rhombus are perpendicular bisectors of each other.
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Question 84 Marks
Construct a quadrilateral ABCD in which AB = 3.6cm, BC = 3.3cm, AD = 2.7cm, diagonal AC = 4.6cm and diagonal BD = 4cm.
Answer

Steps of construction:
Step 1: Draw AB = 3.6cm.
Step 2: With B as the centre and radius equal to 4cm, draw an arc.
Step 3: With A as the centre and radius equal to 2.7cm, draw another arc, cutting the previous arc at D.
Step 4: Join BD and AD.
Step 5: With A as the centre and radius equal to 4.6cm, draw an arc.
Step 6: With B as the centre and radius equal to 3.3cm, draw another arc, cutting the previous arc at C.
Step 7: Join AC, BC and CD.
Thus, ABCD is the required quadrilateral.
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Question 94 Marks
Construct a quadrilateral ABCD in which AB = 3.5cm, BC = 3.8cm, CD = DA = 4.5cm and diagonal BD = 5.6cm.
Answer

Steps of construction:
Step 1: Draw AB = 3.5cm
Step 2: With B as the centre and radius equal to 5.6cm, draw an arc.
Step 3: With A as the centre and radius equal to 4.5cm, draw another arc, cutting the previous arc at D.
Step 4: Join BD and AD.
Step 5: With D as the centre and radius equal to 4.5cm, draw an arc.
Step 6: With B as the centre and radius equal to 3.8cm, draw another arc, cutting the previous arc at C.
Step 7: Join BC and CD.
Thus, ABCD is the required quadrilateral.
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Question 104 Marks
Construct a quadrilateral ABCD in which AB = BC = 3.5cm, AD = CD = 5.2cm and $\angle\text{ABC}=120^\circ.$
Answer

Steps of construction:
Step 1: Draw AB = 3.5cm.
Step 2: Make $\angle\text{ABC}=120^\circ.$
Step 3: With B as the centre, draw an arc 3.5cm and name that point C.
Step 4: With C as the centre, draw an arc 5.2cm.
Step 5: With A as the centre, draw another arc ​5.2cm, cutting the previous arc at D.
Step 6: Join CD and AD.
Thus, ABCD is the required quadrilateral.
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Question 114 Marks
Construct a rhombus ABCD in which AB = 4cm and diagonal AC is 6.5cm.
Answer

Steps of construction:
Step 1: Draw AB = 4cm.
Step 2: With B as the centre, draw an arc of 4cm.
Step 3: With A as the centre, draw another arc of 6.5cm, cutting the previous arc at C.
​Step 4: Join AC and BC.
Step 5: With C as the centre, draw an arc of 4cm.
Step 6: ​With A as the centre, draw another arc of 4cm, cutting the previous arc at D.
Step 7: Join AD and CD.
ABCD is the required rhombus.
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Question 124 Marks
Construct a square, each of whose sides measures 6.4cm.
Answer
All the sides of a square are equal.
Steps of construction: Step 1: Draw AB = 6.4cm. Step 2: Make $\angle\text{A}=90^\circ$ $\angle\text{B}=90^\circ$ Step 3: Draw an arc of length 6.4cm from point A and name that point as D. Step 4: Draw an arc of length 6.4cm from point B and name that point as C. Step 5: Join C and D. ​Thus, ABCD is a required square.
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Question 134 Marks
Construct a quadrilateral PQRS in which PQ = 4.2cm, $\angle\text{PQR}=60^\circ,\angle\text{QPS}=120^\circ,$ QR = 5cm and PS = 6cm.
Answer
Steps of construction:
Step 1: Take PQ = 4.2cm
Step 2: Make $\angle\text{XPQ}=120^\circ,\angle\text{YQP}=60^\circ$
Step 3: Cut an arc of length 5cm from point Q. Name that point as R.
Step 4: From P, make an arc of length 6cm. Name that point as S.
Step 5: Join P and S.
Thus, PQRS is a quadrilateral.
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Question 144 Marks
Construct a quadrilateral PQRS in which QR = 7.5cm, PR = PS = 6cm, RS = 5cm and QS = 10cm. Measure the fourth side.
Answer

Steps of construction:
Step 1: Draw QR = 7.5cm.
Step 2: With Q as the centre and radius equal to 10cm, draw an arc.
Step 3: With R as the centre and radius equal to 5cm, draw another arc, cutting the previous arc at S.
Step 4: Join QS and RS.
Step 5: With S as the centre and radius equal to 6cm, draw an arc.
Step 6: With R as the centre and radius equal to 6cm, draw another arc, cutting the previous arc at P.
Step 7: Join PS and PR.
Step 8: PQ = 4.9cm.
Thus, PQRS is the required quadrilateral.
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Question 154 Marks
Construct a quadrilateral ABCD in which AB = 2.9cm, BC = 3.2cm, CD = 2.7cm, DA = 3.4cm and $\angle\text{A}=70^\circ.$
Answer

Steps of construction:
Step 1: Draw AB = 2.9cm.
Step 2: Make $\angle\text{A}=70^\circ$
Step 3: With A as the centre, draw an arc of 3.4cm. Name that point as D.
Step 4: With D as the centre, draw an arc of 2.7cm.
Step 5: With B as the centre, draw an arc of 3.2cm, cutting the previous arc at C.
Step 6: Join CD and BC.
Then, ABCD is the required quadrilateral.
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Question 164 Marks
Construct a parallelogram ABCD in which AB = 5.2cm, BC = 4.7cm and AC = 7.6cm.
Answer

Steps of construction:
Step 1: Draw AB = 5.2cm.
Step 2: With B as the centre, draw an arc of 4.7cm.
Step 3: With A as the centre, draw another arc of 7.6cm, cutting the previous arc at C.
Step 4: Join A and C.
Step 5: We know that the opposite sides of a parallelogram are equal. Thus, with C as the centre, draw an arc of 5.2cm.
Step 6: With A as the centre, draw another arc of 4.7cm, cutting the previous arc at D.
Step 7: Join CD and AD.
Then, ABCD is the required parallelogram.
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Question 174 Marks
Construct a parallelogram ABCD in which BC = 5cm, $\angle\text{BCD}=120^\circ$ and CD = 4.8cm.
Answer

Steps of construction:
Step 1: Draw BC = 5cm.
Step 2: Make an $\angle\text{BCD}=120^\circ.$
Step 2: With C as centre draw an arc of 4.8cm, name that point as D.
Step 3: With D as centre draw an arc 5cm, name that point as A.
Step 4: With B as centre draw another arc 4.8cm cutting the previous arc at A.
Step 5: Join AD and AB.
Then, ABCD is a required parallelogram.
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Question 184 Marks
Construct a quadrilateral ABCD in which AB = 4.2cm, BC = 6cm, CD = 5.2cm, DA = 5cm and AC = 8cm.
Answer

Steps of construction:
Step 1: Draw AB = 4.2cm.
Step 2: With A as the centre and radius equal to 8cm, draw an arc.
Step 3: With B as the centre and radius equal to 6cm, draw another arc, cutting the previous arc at C.
Step 4: Join BC.
Step 5: With A as the centre and radius equal to 5cm, draw an arc.
Step 6: With C as the centre and radius equal to 5.2cm, draw another arc, cutting the previous arc at D.
Step 7: Join AD and CD.
Thus, ABCD is the required quadrilateral.
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Question 194 Marks
Construct a quadrilateral ABCD in which AB = 3.5cm, BC = 5cm, CD = 4.6cm, $\angle\text{B}=125^\circ$ and $\angle\text{C}=60^\circ.$
Answer

Steps of construction:
Step 1: Draw BC = 5cm.
Step 2: Make $\angle\text{B}=125^\circ$ and $\angle\text{C}=60^\circ.$
Step 3: With B as the centre, draw an arc of 3.5cm. Name that point as A.
Step 4: With C as the centre, draw an arc of 4.6cm. Name that point as D.
Step 5: Join A and D.
Then, ABCD is the required quadrilateral.
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Question 204 Marks
Construct a parallelogram ABCD in which AB = 6.5cm, AC = 3.4cm and the altitude AL from A is 2.5cm. Draw the altitude from C and measure lt.
Answer

Steps of construction:
Step 1: Draw AB = 6.5cm.
Step 2: Draw a perpendicular at point A. Name that ray as AX. From point A, draw an arc of length 2.5cm, on the ray AX and name that point as L.
Step 3: On point L, make a perpendicular. Draw a straight line YZ passing through L, which is perpandicular to the ray AX.
Step 4: Cut an arc of length 3.4cm on the line YZ and name it as C.
Step 5: From point C, cut an arc of length 6.5cm on the line YZ. Name that point as D.
Step 6: Join BC and AD.
Therefore, quadrilateral ABCD is a parallelogram.
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Question 214 Marks
Construct a parallelogram ABCD in which AB = 4.3cm, AD = 4cm and BD = 6.8cm.
Answer

Steps of construction:
Step 1: Draw AB = 4.3cm.
Step 2: With B as the centre, draw an arc of 6.8cm.
Step 3: With A as the centre, draw another arc of 4cm, cutting the previous arc at D.
Step 4: Join BD and AD.
Step 5: We know that the opposite sides of a parallelogram are equal.
Thus, with D as the centre, draw an arc of 4.3cm.
Step 6: With B as the centre, draw another arc of 4cm, cutting the previous arc at C.
Step 7: Join CD and BC.
Then, ABCD is the required parallelogram.
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