RD Sharma solution class 8 chapter 18 Practical Geometry(constructions) Exercise 18.5

Exercise 18.5

Page-18.13

Question 1:

Construct a quadrilateral ABCD given that AB = 4 cm, BC = 3 cm, ∠A = 75°, ∠B = 80° and ∠C = 120°.

Answer 1:

$$\begin{gathered} \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{AB}=4 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{Construct} \angle \mathrm{XAB}=75° \mathrm{at} \mathrm{A} \mathrm{and} \angle \mathrm{ABY}=80° \mathrm{at} \mathrm{B}. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{B} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 3 \mathrm{cm}, \mathrm{cut} \mathrm{off} \mathrm{BC}=3 \mathrm{cm}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{At} \mathrm{C}, \mathrm{draw} \angle \mathrm{BCZ}=120° \mathrm{such} \mathrm{that} \mathrm{it} \mathrm{meets} \mathrm{AX} \mathrm{at} \mathrm{D}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 2:

Construct a quadrilateral ABCD, where AB = 5.5 cm, BC = 3.7 cm, ∠A = 60°, ∠B = 105° and ∠D = 90°.

Answer 2:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{all} \mathrm{the} \mathrm{angles} \mathrm{in} \mathrm{a} \mathrm{quadrilateral} \mathrm{is} 360. \\ \mathrm{i}.\mathrm{e}., \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}+\angle \mathrm{D}=360° \\ \Rightarrow \angle \mathrm{C}=105° \\ \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{AB}=5.5 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{Construct} \angle \mathrm{XAB}=60° \mathrm{at} \mathrm{A} \mathrm{and} \angle \mathrm{ABY}=105°. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{B} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 3.7 \mathrm{cm}, \mathrm{cut} \mathrm{off} \mathrm{BC}=3.7 \mathrm{cm}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{At} \mathrm{C}, \mathrm{draw} \angle \mathrm{BCZ}=105° \mathrm{such} \mathrm{that} \mathrm{it} \mathrm{meets} \mathrm{AX} \mathrm{at} \mathrm{D}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 3:

Construct a quadrilateral PQRS, where PQ = 3.5 cm, QR = 6.5 cm, ∠P = ∠R = 105° and ∠S = 75°.

Answer 3:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{all} \mathrm{the} \mathrm{angles} \mathrm{in} \mathrm{a} \mathrm{quadrilateral} \mathrm{is} 360. \\ \mathrm{i}.\mathrm{e}., \angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{R}+\angle \mathrm{S}=360° \\ \Rightarrow \angle \mathrm{Q}=75° \\ \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{PQ}=3.5 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{Construct} \angle \mathrm{XPQ}=105° \mathrm{at} \mathrm{P} \mathrm{and} \angle \mathrm{PQY}=75° \mathrm{at} \mathrm{Q}. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{Q} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 6.5 \mathrm{cm}, \mathrm{cut} \mathrm{off} \mathrm{QR}=6.5 \\ \mathrm{Step} \mathrm{IV}: \mathrm{At} \mathrm{R}, \mathrm{draw} \angle \mathrm{QRZ}=105° \mathrm{such} \mathrm{that} \mathrm{it} \mathrm{meets} \mathrm{PX} \mathrm{at} \mathrm{S}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 4:

Construct a quadrilateral ABCD when BC = 5.5 cm, CD = 4.1 cm, ∠A = 70°, ∠B = 110° and ∠D = 85°.

Answer 4:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{all} \mathrm{the} \mathrm{angles} \mathrm{in} \mathrm{a} \mathrm{quadrilateral} \mathrm{is} 360. \\ \mathrm{i}.\mathrm{e}., \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}+\angle \mathrm{D}=360° \\ \Rightarrow \angle \mathrm{C}=95° \\ \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{BC}=5.5 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{Construct} \angle \mathrm{XBC}=110° \mathrm{at} \mathrm{A} \mathrm{and} \angle \mathrm{BCY}=95°. \\ \mathrm{Step} \mathrm{III}: \mathrm{With} \mathrm{C} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 4.1 \mathrm{cm}, \mathrm{cut} \mathrm{off} \mathrm{CD}=4.1 \mathrm{cm}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{At} \mathrm{D}, \mathrm{draw} \angle \mathrm{CDZ}=85° \mathrm{such} \mathrm{that} \mathrm{it} \mathrm{meets} \mathrm{BY} \mathrm{at} \mathrm{A}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 5:

Construct a quadrilateral ABCD, where ∠A = 65°, ∠B = 105°, ∠C = 75° BC = 5.7 cm and CD = 6.8 cm.

Answer 5:

$$\begin{gathered} \mathrm{We} \mathrm{know} \mathrm{that} \mathrm{the} \mathrm{sum} \mathrm{of} \mathrm{all} \mathrm{the} \mathrm{angles} \mathrm{in} \mathrm{a} \mathrm{quadrilateral} \mathrm{is} 360. \\ \mathrm{i}.\mathrm{e}., \angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}+\angle \mathrm{D}=360° \\ \Rightarrow \angle \mathrm{D}=115° \\ \mathrm{Steps} \mathrm{of} \mathrm{Construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{BC}=5.7 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{Construct} \angle \mathrm{XBC}=105° \mathrm{at} \mathrm{B} \mathrm{and} \angle \mathrm{BCY}=105° \mathrm{at} \mathrm{C}. \\ \mathrm{Step} \mathrm{III} : \mathrm{With} \mathrm{C} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 6.8 \mathrm{cm}, \mathrm{cut} \mathrm{off} \mathrm{CD}=6.8 \mathrm{cm}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{At} \mathrm{D}, \mathrm{draw} \angle \mathrm{CDZ}=115° \mathrm{such} \mathrm{that} \mathrm{it} \mathrm{meets} \mathrm{BY} \mathrm{at} \mathrm{A}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$

Question 6:

Construct a quadrilateral PQRS, in which PQ = 4 cm, QR = 5 cm, ∠P = 50°, ∠Q = 110° and ∠R = 70°.

Answer 6:

$$\begin{gathered} \mathrm{Steps} \mathrm{of} \mathrm{construction}: \\ \mathrm{Step} \mathrm{I}: \mathrm{Draw} \mathrm{PQ}=4 \mathrm{cm}. \\ \mathrm{Step} \mathrm{II}: \mathrm{Construct} \angle \mathrm{XPQ}=50° \mathrm{at} \mathrm{P} \mathrm{and} \angle \mathrm{PQY}=110° \mathrm{at} \mathrm{Q}. \\ \mathrm{Step} \mathrm{III} : \mathrm{With} \mathrm{Q} \mathrm{as} \mathrm{the} \mathrm{centre} \mathrm{and} \mathrm{radius} 5 \mathrm{cm}, \mathrm{cut} \mathrm{off} \mathrm{QR}=5 \mathrm{cm}. \\ \mathrm{Step} \mathrm{IV}: \mathrm{At} \mathrm{R}, \mathrm{draw} \angle \mathrm{QRZ}=70° \mathrm{such} \mathrm{that} \mathrm{it} \mathrm{meets} \mathrm{PX} \mathrm{at} \mathrm{S}. \\ \mathrm{The} \mathrm{quadrilateral} \mathrm{so} \mathrm{obtained} \mathrm{is} \mathrm{the} \mathrm{required} \mathrm{quadrilateral}. \end{gathered}$$