Constructions Of Triangles
Class 9th Mathematics Part Ii MHB Solution
- Construct Δ PQR, in which QR= 4.2cm, ∠Q = 40° and PQ+PR= 8.5cm
- Construct ΔXYZ, in which YZ= 6cm, XY+XZ= 9cm and ∠Y = 50°
- Construct Δ ABC, in which BC= 6.2cm, ∠C = 50°, AB+AC= 9.8cm
- Construct ΔABC, in which BC= 5.2cm, ∠C = 45° and perimeter of ΔABC is 10 cm.…
- Construct Δ XYZ, in which YZ= 7.4cm, ∠XYZ = 45° and XY-XZ= 2.7cm
- Construct Δ PQR, in which QR = 6.5cm, ∠PQR = 60° and PQ-PR = 2.5cm…
- Construct Δ ABC, in which BC = 6cm, ∠ABC = 100° and AC - AB - 2.5CM…
Practice Set 4.1
Question 1.Construct Δ PQR, in which QR= 4.2cm, ∠Q = 40° and PQ+PR= 8.5cm
Answer:
Given: base QR= 4.2cm, PQ+PR= 8.5cm and
∠Q = 40° of Δ PQR.
Required: To construct a ΔPQR
Steps of construction:
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i: Draw a segment QR of length of 4.2cm.
ii: At vertex Q, construct ∠Q = 40° and produce a ray QB.
iii: Mark an arc on ray QB cutting at D such that QD = 8.5cm.
iv: Draw segment DR.
v: Construct the perpendicular bisector of segment DR.
vi: Name the point of intersection of ray QB and the perpendicular bisector of DR as P.
vii: Draw segment RP.
ΔPQR is the required triangle.
Question 2.
Construct ΔXYZ, in which YZ= 6cm, XY+XZ= 9cm and ∠Y = 50°
Answer:
Given:
base YZ= 6cm,
XY+XZ= 9cm and
∠Y = 50°
Required: To construct a ΔXYZ
Steps of construction:
i: Draw a segment YZ of length of 6cm.
ii: At vertex Y, construct ∠Y = 50° and produce a ray YA.
iii: Mark an arc on ray YA cutting at D such that YD = 9cm.
iv: Draw segment ZD.
v: Construct the perpendicular bisector of segment ZD.
vi: Name the point of intersection of ray YD and the perpendicular bisector of ZD as X.
vii: Draw segment XZ.
ΔXYZ is the required triangle.
Question 3.
Construct Δ ABC, in which BC= 6.2cm, ∠C = 50°, AB+AC= 9.8cm
Answer:
Given: base BC= 6.2cm,
AB+AC= 9.8cm and
∠C = 50° of Δ ABC.
Required: To construct a ΔABC
Steps of construction:
i: Draw a segment BC of length of 6.2cm.
ii: At vertex C, construct ∠C = 50° and produce a ray CP.
iii: Mark an arc on ray CP cutting at D such that CD = 9.8cm.
iv: Draw segment DB.
v: Construct the perpendicular bisector of segment DB.
vi: Name the point of intersection of ray CP and the perpendicular bisector of DB as A.
vii: Draw segment AB.
ΔABC is the required triangle.
Question 4.
Construct ΔABC, in which BC= 5.2cm, ∠C = 45° and perimeter of ΔABC is 10 cm.
Answer:
Given: base BC= 5.2cm, ∠C = 45° of ΔABC and
perimeter of ΔABC = 10 cm.
Required: To construct a ΔABC
Steps of construction:
i: Draw a segment BC of length of 5.2cm.
ii: At vertex C, construct ∠C = 45° and produce a ray P’C
iii: Mark an arc on ray CP cutting at D such that CD = 4.8cm.
iv: Draw segment DB.
v: Construct the perpendicular bisector of segment DB.
vi: Name the point of intersection of ray CP and the perpendicular bisector of DB as A.
vii: Draw segment AB.
ΔABC is the required triangle.
Practice Set 4.2
Question 1.Construct Δ XYZ, in which YZ= 7.4cm, ∠XYZ = 45° and XY-XZ= 2.7cm
Answer:
Given: Base YZ= 7.4cm, XY-XZ= 2.7cm and
∠Y = 45° of Δ XYZ.
Required: To construct a triangle XYZ.
Steps of construction:
i: Draw a segment YZ of length 7.4cm.
ii: Draw ray YL such that ∠Y = 45°
iii: Mark an arc on ray YL cutting at D such that YD = 2.7cm.
iv: Draw segment ZD.
v: Construct the perpendicular bisector of segment ZD.
vi: Name the point of intersection of ray YL and the perpendicular bisector of ZD as X.
vii: Draw segment XZ.
ΔXYZ is the required triangle.
Question 2.
Construct Δ PQR, in which QR = 6.5cm, ∠PQR = 60° and PQ-PR = 2.5cm
Answer:
Given: Base QR = 6.5cm,
PQ-PR = 2.5cm and
∠Q = 60° of Δ PQR.
Required: To construct a triangle PQR.
Steps of construction:
i: Draw a segment QR of length 6.5cm.
ii: Draw ray QL such that ∠Q = 60°
iii: Mark an arc on opposite ray QL i.e. QS cutting at D such that QD = 2.5cm.
iv: Draw segment RD.
v: Construct the perpendicular bisector of segment RD.
vi: Name the point of intersection of ray QL and the perpendicular bisector of RD as P.
vii: Draw segment PR.
ΔPQR is the required triangle.
Question 3.
Construct Δ ABC, in which BC = 6cm, ∠ABC = 100° and AC – AB – 2.5CM
Answer:
Given: Base BC = 6cm, AC-AB = 2.5cm and
∠B = 100° of Δ ABC.
Required: To construct a triangle ABC.
Steps of construction:
i: Draw a segment BC of length 6cm.
ii: Draw ray BT such that ∠B = 100°
iii: Mark an arc on opposite of ray BT i.e. ray BS cutting at D such that BD = 2.5cm.
iv: Draw segment CD.
v: Construct the perpendicular bisector of segment CD.
vi: Name the point of intersection of ray BS and the perpendicular bisector of CD as A.
vii: Draw segment AC.
ΔABC is the required triangle.
Practice Set 4.3
Question 1.Construct Δ PQR, in which ∠Q =70°, ∠R = 80° and PQ+QR+PR=9.5cm
Answer:
Given: ∠Q =70°,
∠R = 80° and
perimeter of Δ PQR
Steps of construction:
i: Draw a line segment XY of 9.5cm.
ii: From point X draw a ray XD at 70° and from Y draw a ray YE at 80°.
iii: Draw an angle bisector of X and Y, two angle bisectors intersect each other at point A.
iv: Draw a line bisector of XA and AY respectively these two-line bisectors intersect at point B and C.
v: Join AB and AC.
vi: Δ ABC is required triangle.
Question 2.
Construct ΔXYZ in which ∠Y =58°, ∠X = 46° and perimeter of triangle is 10.5 cm.
Answer:
Given: ∠Y =58°, ∠X = 46° and perimeter of Δ PQR
Steps of construction:
i: Draw a line segment QR of 10.5cm.
ii: From point Q draw a ray QD at 58° and from R draw a ray RE at 46°
iii: Draw an angle bisector of Q and R, two angle bisectors intersect each other at point X.
iv: Draw a line bisector of QX and XR respectively these two-line bisectors intersect at point Y and Z
v: Join XY AND XZ.
vi: Δ XYZ is required triangle.
Question 3.
Construct ΔLMN, in which ∠M =60°, ∠N =80° and LM+MN+NL=11cm.
Answer:
Given: ∠Q =70°,
∠R = 80° and
perimeter of Δ PQR
Steps of construction:
i: Draw a line segment AB of 11cm.
ii: From point A draw a ray AD at 60° and from B draw a ray BE at 80°
iii: Draw an angle bisector of A and B, two angle bisectors intersect each other at point P.
iv: Draw a line bisector of AP and BP respectively these two-line bisectors intersect at point Q and R
v: Join PQ AND PR
vi: Δ PQR is required triangle.
Problem Set 4
Question 1.Construct ΔXYZ, such that XY+XZ= 10.3cm, YZ=4.9cm, ∠XYZ = 45°
Answer:
Given: base YZ= 6cm, XY+XZ= 9cm and ∠Y = 50°
of ΔXYZ.
Required: To construct a ΔXYZ
Steps of construction:
i: Draw a segment YZ of length of 4.9cm.
ii: At vertex Y, construct ∠Y = 45° and produce a ray YA.
iii: Mark an arc on ray YA cutting at D such that YD = 9cm.
iv: Draw segment ZD.
v: Construct the perpendicular bisector of segment ZD.
vi: Name the point of intersection of ray YD and the perpendicular bisector of ZD as X.
vii: Draw segment XZ.
ΔXYZ is the required triangle.
Question 2.
Construct Δ ABC, in which ∠B =70°, ∠C = 60°, AB+BC+AC=11.2cm.
Answer:
Given: ∠B =70°, ∠C = 60° and perimeter of Δ ABC
Steps of construction:
i: Draw a line segment PQ of 11.2cm.
ii: From point P draw a ray PD at 70° and from Q draw a ray QE at 60°
iii: Draw an angle bisector of P and Q, two angle bisectors intersect each other at point A.
iv: Draw a line bisector of AP and AQ respectively these two-line bisectors intersect at point B and C
v: Join AB AND AC
vi: Δ ABC is required triangle.
Question 3.
The perimeter of a triangle is 14.4 cm and the ratio of lengths of its side is 2: 3: 4. Construct the triangle.
Answer:
Given: perimeter of Δ = 14.4cm and ratio of its sides = 2:3:4
Steps of construction:
i: Draw B’C’ = 9cm
ii: Then divide B’C’ in the ratio 2:3:4. B’B:BC∷CC’ = 2:3:4 (by similarity of triangles)
iii: Now construct AB = B’B.
iv: Construct AC = CC’.
v: Hence, AB:BC:AC = 2:3:4
vi: Hence, Δ ABC is the required triangle.
Question 4.
Construct Δ PQR, in which PQ-PR= 2.4cm, QR= 6.4cm and ∠PQR = 55°.
Answer:
Given: Base QR= 6.4cm, PQ-PR= 2.4cm and
∠Q = 545° of Δ PQR.
Required: To construct a triangle PQR.
Steps of construction:
i: Draw a segment QR of length 6.4cm.
ii: Draw ray QL such that ∠Q = 55°
iii: Mark an arc on ray QL cutting at D such that QD = 2.4cm.
iv: Draw segment RD.
v: Construct the perpendicular bisector of segment RD.
vi: Name the point of intersection of ray QL and the perpendicular bisector of RD as P.
vii: Draw segment PR.
ΔPQR is the required triangle.