RD Sharma solution class 8 chapter 18 Practical Geometry(constructions) Exercise 18.1

Exercise 18.1

Page-18.4

Question 1:

Construct a quadrilateral ABCD in which AB = 4.4 cm, BC = 4 cm, CD = 6.4 cm, DA = 3.8 cm and BD = 6.6 cm.

Answer 1:

$$\begin{gathered} \mathrm{First}, \mathrm{we} \mathrm{draw} \mathrm{a} \mathrm{rough} \mathrm{sketch} \mathrm{of} \mathrm{the} \mathrm{quadrilateral} \mathrm{ABCD} \mathrm{and} \mathrm{write} \mathrm{down} \mathrm{its} \mathrm{dimensions} \mathrm{along} \mathrm{the} \mathrm{sides}. \\ \mathrm{We} \mathrm{may} \mathrm{divide} \mathrm{the} \mathrm{quadrilateral} \mathrm{into} \mathrm{two} \mathrm{constructible} \mathrm{triangles} \mathrm{ABD} \mathrm{and} \mathrm{BCD}. \\ \mathrm{Steps} \mathrm{of} \mathrm{Construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{BD}=6.6 \mathrm{cm} \\ \mathrm{Step} \mathrm{II}: \mathrm{With} \mathrm{B} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} \mathrm{BC}=4 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc}. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{D} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 6.4 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{II} \mathrm{at} \mathrm{C}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{With} \mathrm{B} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 4.4 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{on} \mathrm{the} \mathrm{side} \mathrm{BD} \mathrm{opposite} \mathrm{to} \mathrm{that} \mathrm{of} \mathrm{C}. \\ \mathrm{Step} \mathrm{V}: \mathrm{With} \mathrm{D} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 3.8 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{IV} \mathrm{at} \mathrm{A}. \\ \mathrm{Step} \mathrm{VI}: \mathrm{Join} \mathrm{BA}, \mathrm{DA}, \mathrm{BC} \mathrm{and} \mathrm{CD} \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{ABCD} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 2:

Construct a quadrilateral ABCD such that AB = BC = 5.5 cm, CD = 4 cm, DA = 6.3 cm and AC = 9.4 cm. Measure BD.

Answer 2:

$$\begin{gathered} \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{AB}=5.5 \mathrm{cm} \\ \mathrm{Step} \mathrm{II}: \mathrm{With} \mathrm{B} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} \mathrm{BC}=5.5 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc}. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{A} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} \mathrm{AC}=9.4 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{II} \mathrm{at} \mathrm{C}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{With} \mathrm{C} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} \mathrm{CD}=4 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc}. \\ \mathrm{Step} \mathrm{V}: \mathrm{With} \mathrm{A} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} \mathrm{AD}=6.3 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{IV} \mathrm{at} \mathrm{D}. \\ \mathrm{Step} \mathrm{VI}: \mathrm{Join} \mathrm{DA}, \mathrm{BC}, \mathrm{AC} \mathrm{and} \mathrm{CD}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{ABCD} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 3:

Construct a quadrilateral XYZW in which XY = 5 cm, YZ = 6 cm, ZW = 7 cm, WX = 3 cm and XZ = 9 cm.

Answer 3:

$$\begin{gathered} \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{XZ}=9 \mathrm{cm} \\ \mathrm{Step} \mathrm{II}: \mathrm{With} \mathrm{X} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 5 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{above} \mathrm{XZ}. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{Z} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 6 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{II} \mathrm{at} \mathrm{Y} \mathrm{above} \mathrm{XZ}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{With} \mathrm{Z} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 7 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{below} \mathrm{XZ}. \\ \mathrm{Step} \mathrm{V}: \mathrm{With} \mathrm{X} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 3 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{IV} \mathrm{at} \mathrm{W} \mathrm{below} \mathrm{XZ}. \\ \mathrm{Step} \mathrm{VI}: \mathrm{Join} \mathrm{XY}, \mathrm{YZ}, \mathrm{ZW} \mathrm{and} \mathrm{XW}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{WXYZ} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 4:

Construct a parallelogram PQRS such that PQ = 5.2 cm, PR = 6.8 cm and QS = 8.2 cm.

Answer 4:

$$\begin{gathered} \mathrm{In} \mathrm{a} \mathrm{parallelogram} \mathrm{opposite} \mathrm{sides} \mathrm{are} \mathrm{equal}. \mathrm{Thus}, \mathrm{we} \mathrm{have} \mathrm{to} \mathrm{construct} \mathrm{a} \mathrm{quadrilateral} \mathrm{PQRS} \mathrm{in} \mathrm{which} \\ \mathrm{PQ}=5.2 \mathrm{cm}, \mathrm{PR}=6.8 \mathrm{cm} \mathrm{and} \mathrm{QS}=8.2 \mathrm{cm}. \\ \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{QS}=8.2 \mathrm{cm} \\ \mathrm{Step} \mathrm{II}: \mathrm{With} \mathrm{Q} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 5.2 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc}. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{S} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 5.2 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{II} \mathrm{at} \mathrm{C}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{With} \mathrm{P} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 6.8 \mathrm{cm}. \\ \mathrm{Step} \mathrm{V}: \mathrm{With} \mathrm{Q} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 5.2 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{Step} \mathrm{IV} \mathrm{at} \mathrm{A}. \\ \mathrm{Step} \mathrm{VI}: \mathrm{Join} \mathrm{QR}, \mathrm{QP}, \mathrm{PS} \mathrm{and} \mathrm{SR}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{PQRS} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 5:

Construct a rhombus with side 6 cm and one diagonal 8 cm. Measure the other diagonal.

Answer 5:

$$\begin{gathered} \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} 1: \mathrm{Draw} \mathrm{AC}=8 \mathrm{cm}. \\ \mathrm{Step} 2: \mathrm{With} \mathrm{A} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius}=6 \mathrm{cm}, \mathrm{draw} \mathrm{arcs} \mathrm{on} \mathrm{both} \mathrm{sides}. \\ \mathrm{Step} 3: \mathrm{With} \mathrm{C} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius}=6 \mathrm{cm}, \mathrm{draw} \mathrm{arcs} \mathrm{on} \mathrm{both} \mathrm{sides}, \mathrm{intersecting} \mathrm{the} \mathrm{previous} \mathrm{arcs} \mathrm{at} \mathrm{points} \mathrm{B} \mathrm{and} \mathrm{D}. \\ \mathrm{Step} 4: \mathrm{Join} \mathrm{BD} =8.9\mathrm{cm}. \\ \mathrm{Thus}, \mathrm{ABCD} \mathrm{is} \mathrm{t}\mathrm{he} \mathrm{required} \mathrm{rhombus}. \end{gathered}$$

Question 6:

Construct a kite ABCD in which AB = 4 cm, BC = 4.9 cm and AC = 7.2 cm.

Answer 6:

$$\begin{gathered} \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{AC}=7.2 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{With} \mathrm{A} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 4\mathrm{cm}, \mathrm{draw} \mathrm{arcs} \mathrm{on} \mathrm{both} \mathrm{sides} \mathrm{of} \mathrm{the} \mathrm{line} \mathrm{segment} \mathrm{AC}. \\ \mathrm{Step} \mathrm{III} : \mathrm{With} \mathrm{C} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 4.9 \mathrm{cm}, \mathrm{draw} \mathrm{arcs} \mathrm{on} \mathrm{both} \mathrm{sides} \mathrm{of} \mathrm{AC} \mathrm{intersecting} \mathrm{the} \mathrm{previous} \mathrm{arcs} \mathrm{of} \mathrm{step} \mathrm{II} \mathrm{at} \mathrm{B} \mathrm{and} \mathrm{D}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{Join} \mathrm{BA}, \mathrm{DA}, \mathrm{BC} \mathrm{and} \mathrm{CD}. \\ \mathrm{Thus}, \mathrm{the} \mathrm{quadrilateral} \mathrm{ABCD} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{kite}. \end{gathered}$$

Question 7:

Construct, if possible, a quadrilateral ABCD given AB = 6 cm, BC = 3.7 cm, CD = 5.7 cm, AD = 5.5 cm and BD = 6.1 cm. Give reasons for not being able to construct it, if you cannot.

Answer 7:

$$\begin{gathered} \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{AB}=6 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{With} \mathrm{A} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 5.5 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc}. \\ \mathrm{Step} \mathrm{III} : \mathrm{With} \mathrm{B} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 6.1 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{StepII} \mathrm{at} \mathrm{D}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{With} \mathrm{B} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 3.7 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{on} \mathrm{the} \mathrm{side}. \\ \mathrm{Step} V :\mathrm{With} \mathrm{D} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 5.7 \mathrm{cm}, \mathrm{draw} \mathrm{an} \mathrm{arc} \mathrm{to} \mathrm{intersect} \mathrm{the} \mathrm{arc} \mathrm{drawn} \mathrm{in} \mathrm{StepIV} \mathrm{at} \mathrm{C}. \\ \mathrm{Step} \mathrm{VI}: \mathrm{Join} \mathrm{BD}, \mathrm{DA}, \mathrm{BC} \mathrm{and} \mathrm{CD}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{ABCD} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 8:

Construct, if possible, a quadrilateral ABCD in which AB = 6 cm, BC = 7 cm, CD = 3 cm, AD = 5.5 cm and AC = 11 cm. Give reasons for not being able to construct, if you cannot. (Not possible, because in triangle ACD, AD + CD < AC).

Answer 8:

Such a quadrilateral cannot be constructed because in a triangle, the sum of the length of its two sides must be greater than the that of the third side

But here in triangle ACD,
AD + CD = 5.5 + 3 = 8.5 cm
and AC = 11 cm

i.e., AD + CD <  AC, which is not possible.

So, the construction is not possible.