CONSTRUCTION OF TRIANGLE

POINTS TO REMEMBER

Perpendicular bisector : The line which is perpendicular to the given line segment and also

bisects the line segment.

Angle bisector : The line which divides a complete angle into two equal angles.

Perimeter : Sum of all the sides of given geometrical figure or the length of the boundary of

any plane figure.

Altitude : A perpendicular drawn from the vertex to the opposite side.

Arc : Part of the circle.

Base angle : Any of the angles formed by the base of a triangle, with the other sides.

Vertex angle : The angle at the top of an isosceles triangle.

Median : the line drawn from the vertex to the mid point of the opposite side.

Note : 1) Atleast three parameters are needed to construct a triangle. 2) Not all combination of

three parameters will enable one to construct a triangle.

EXERCISE 3.4.2

1. Construct a triangle ABC in which AB=5cm and BC=4.6cm and AC=3.7cm.

c

3.7cm 4.5cm

A 5 cm B

2. Construct an equilateral triangle of side 4.8cm.

L

4.8cm 4.8cm

J K

4.8cm

3. Construct a triangle PQR, given that PQ=5.6, PR=7cm and QR=4.5cm.

Q

5.6cm 4.5cm

P 7 cm R

4. Construct a triangle XYZ in which XY=7.8cm, YZ=4.5cm and XZ=9.5cm.

Y

7.8cm

4.5cm

X 7 cm Z

5. Construct a triangle whose perimeter is 12cm and the ratio of their sides is 3:4:5.

Perimeter=12cm

Ratio=3:4:5

Sum of terms of ratio=3+4+5=12

Length of 1st side =

3

12∗ 12 = 3cm

Length of 2nd

side =4

12∗ 12 = 4cm

Length of 2nd

side =5

12∗ 12 = 5cm

K

3cm 4cm

J L

5 cm

EXERCISE 3.4.3

6. Construct a triangle ABC, in which AB=4.5, AC=5.5cm and BAC=750.

[Hint: Since, A=750 , A should be present in the base line]

B

4.5cm

750

A C 5.5cm

2. Construct a triangle PQR in which PQ=5.4cm and PQR=550.

P

5.4cm

Q 550 R

5.5 cm

3. Construct a triangle XYZ in which XY=5cm, YZ=5.5cm and XYZ=1000.

X

5cm

Y 1000 z

4. Construct a triangle LMN in which LM=7.8cm, MN=6.3 and LMN=450.

N

6.3cm

L 45 M

450

EXERCISE 3.4.4

1. Construct a triangle ABC in which AB=6.5cm, A=450 and B=60

0.

[Hint: Since A and B are given, select the base line as AB] C

450 60

0

A 6.5cm B

2. Construct a triangle PQR in which QR=4.8cm, Q=450 and R=55

0.

p

45

0 55

0

3. Construct a triangle ABC in which BC=5.2 cm, B=350 and C=80

0.

A

350 800

B 5.2cm C

4. Construct a triangle ABC in which BC=6cm, B=300 and C=1250

.

A

300 1250

B 6cm C

EXERCISE 3.4.5

1. Construct an equilateral triangle of height 4.5cm. Measure approximate length of its

sides.

Steps of Construction

i) Draw any line segment XY.

ii) Take any point M on XY. Draw ZM XY.

iii) With M as center and radius 4.5cm, draw an arc cutting MZ at A.

iv) Construct MAB=300 and MAC=30

0, with B and C on XY.

Then ABC is the required triangle.

Z

A

X B M C Y

Approximate 2cm length of side is 5.

2. Construct an equilateral triangle of height 5.2cm. Measure approximate length of its

sides.

Z A

X B M C Y

3. Construct an equilateral triangle of height 5.2cm. Measure approximate length of its

sides.

Z

A

X B M C Y

EXERCISE 3.4.6

1. Construct a triangle PQR in which PQ=5.5cm, PR=6.2cm and length of the perpendicular

from P on QR is 4cm.

Steps of construction

Z

P

X Q M R Y

i) Draw a line segment XY.

ii) Take a point M on XY.

iii) Draw ZM XY, with MZ sufficiently large.

iv) With M as center and radius 4cm, draw an arc cutting MZ at P.

v) With P as center and radius 5.5cm and 6.2 cm draw arcs cutting XY at Q and

R respectively. Join PQ and PR. Then PQR is the required triangle.

2. Construct a triangle MNP in which MN=4.5cm, MP=5.2cm and length of the

perpendicular from M on NP is 3.8cm.

D

M

B N A P C

EXERCISE 3.4.7

1. Construct a right angle triangle ABC in which B=900, AB=5cm and AC=7cm.

P

A

900

X B C Y

Steps of construction

1) Draw a line segment XY.

2) Locate A, B on XY such that AB=5cm.

3) Construct a line segment BP, such that it is sufficiently large and ABP=900.

4) With A as center and radius 7cm draw an arc cutting BP at C. join AC.

5) Then, ABC is the required triangle.

2. Construct a right angle triangle PQR in which R= 900, PQ=4cm, and QR=3cm.

P

900

Q 3cm R

3. Construct a right angle triangle ABC in which B=900, BC=4cm and AC=5cm.

A

900

B C

EXERCISE 3.4.8

1. Construct an isosceles triangle ABC in which base BC=6.5cm and altitude from A on BC

is 4cm.

P

A

B 6.5cm M C

Steps of construction

1) Drew a line segment BC whose length is 6.5cm.

2) Draw the perpendicular bisector of BC, call it MP with M on BC.

3) With M as centre and radius 4cm, draw an arc cutting MP at A, join AB and AC. Then

ABC is the required triangle.

2. Construct an isosceles triangle XYZ in which base YZ=6.5cm and altitude from X on YZ

is 3.8cm.

P

X

Y 6.5cm M Z

3. Construct an isosceles triangle PQR in which base PQ=7.2cm and altitude from R on PQ

is 5cm.

P

R

P 7.2cm M Q

EXERCISE 3.4.9

1. Construct an isosceles triangle whose altitude is 4.5cm and vertex angle is 700.

Steps of Construction:

1) Draw a line segment XY.

2) Take a point M on XY and draw a line MP XY.

3) With M as centre and radius 4.5cm, draw an arc cutting MP at A.

4) B and C on XY such that MAB=70 °

2= 35° and MAC=

70

2= 35°

Then ABC is the required triangle.

P

A

X B M C Y

2. Construct an isosceles triangle whose altitude is 6.6cm and vertex angle is 600.

A A

B M C

3. Construct an isosceles triangle whose altitude is 5cm and vertex angle is 900.

A

X B M C Y

EXERCISE 3.4.10

1. Construct a triangle ABC, whose perimeter is 13cm and whose sides are in the ratio

3:4:5. A 5

3

B 4 C

2. Construct a triangle PQR, whose perimeter is 14cm and whose sides are in the ratio 2:4:5.

A

5 3

B 4 C L

M N

3. Construct a triangle PQR, whose perimeter is 14cm and whose sides are in the ratio 2:4:5.

P

4 2

X Q 3 R Y

L

M N

EXERCISE 3.4.11

1. Construct a triangle ABC whose perimeter 12cm, and whose base angles are 500 and

800.

Steps of Construction: 1) Draw a line segment and locate points P, Q such that PQ=12.5cm.

2) Construct rays PR such that QPR=600 and QS such that PQS=75

0.

3) Draw the perpendicular bisector of AP and AQ and let these intersect PQ at B and

C respectively.

4) Join AB and AC. Then ABC is the required triangle.

R S

M A L

500 80

0

2. Construct a triangle XYZ whose perimeter 15cm. and whose base angles are 600 and

700.

Y

600 70

0

3. Construct a triangle ABC whose base angles are 650 and 85

0.

R S

A

650 85

0

EXERCISE 3.4.12

1. Construct a triangle ABC in which BC=3.6cm, AB+AC=4.8cm and B=600.

X

D

A

B 600 C

Steps of Construction

1) Draw a line segment BC of 3.6cm.

2) Draw a line segment BX, sufficiently large such that XBC= 600.

3) Form the segment BX, cut off line segment BD of length 4.8cm.

4) Join CD.

5) Draw the perpendicular bisector of CD and let it meet BD at A.

6) Join AC.

Then ABC is required triangle.

2. Construct a triangle ABC in which AB+AC=5.6cm, BC=4.5cm and B=450.

X

D

A

B 450 C

3.Construct a triangle PQR in which PQ+PR=6.5cm, QR=5.4cm and Q=400.

X

D

P

Q 400 R

EXERCISE 3.4.13

1. Construct triangle ABC in which BC=3.4cm, AB-AC=1.5cm and B=450.

X

A

D

450

B 3.4cm C

Step of Construction

1. Draw a line segment BC of length 3.4cm. 2. Draw another line segment BX, sufficiently large, such that XBC=45

0.

3. From the segment BX, cut off line segment BD=1.5cm, which is equal to (AB-AC). 4. Join CD.

5. Draw the perpendicular bisector of CD and let it cut BX at A. 6. Join AC.

Then ABC is the required triangle.

2. Construct triangle ABC in which BC=5cm, AB-AC=2.8cm and B=400.

X

A

D

400

B 5cm C

3. Construct triangle ABC in which BC=6cm, AB-AC=3.1cm and B=300.

X

A

D

300

B 6 cm C

ADDITIONAL PROBLEMS ON “CONSTRUCTION OF TRIANGLES”

1. Construct a triangle ABC in which AB=5cm, BC=4.7cm, AC=4.3cm.

C

A 5cm B

2. Construct a triangle ABC in which AB=5cm, BC=5cm and AC=4.3cm.

C

A 5cm B

3.Construct a triangle PQR in which PQ=4cm QR=4.5cm, Q=600.

R

P 4cm Q

3. Construct a triangle PQR in which PQ=4cm, P=600 and Q=60

0.

R

600

P 4cm Q

4. Construct a triangle ABC in which AB=3.5cm, AC=4cm and length of perpendicular

from A to BC is 3cm.

A

3cm

C M B

5.Construct an isosceles triangle ABC in which base BC=4.5cm and altitude from A on

BC is 3.8cm.

A

B 4.5cm C

5. Construct an isosceles triangle whose altitude is 5cm and vertex angle is 700.

A

300 300

5cm

C M B

6. Construct an isosceles triangle whose altitude is 5cm and vertex angle is 800.

A

400 400

5cm

C M B

7. Construct an equilateral triangle of height 3.5cm.

J

300 300

3.5cm

L M K

8. Construct an equilateral triangle of height 4.3cm.

J

300 300

4.5cm

L M K

9. Construct right angle triangle LMN in which M=900, MN=4.5cm, LN=5.6cm.

L

5.6cm

900

M 4.5cm N

10. Construct right angle triangle PQR in which Q=900, QR=4.5cm and R=50

0.

P

900 50

0

Q 4.5cm R

11. Construct a triangle PQR, whose perimeter is 13cm and whose sides are in the ratio

2:3:4.

P

x Q R Y

Y 13cm

M

N

12. Construct a triangle PQR, whose perimeter is 15cm and whose sides are in the ratio

3:4:6.

P

x Q R Y

Y 15cm

M

13. Construct a triangle ABC, whose perimeter is 13.5cm. whose base angels

are 600 and 75

0.

R S

A

600 75

0

P 13.5cm B C Q

14. Construct a triangle ABC, whose perimeter is 12.5cm, and whose base angles are 500

and 800.

R S

A

500 80

0

P 12.5cm B C Q

15. Construct a triangle XYZ in which YZ=4.5cm, Y=600 and sum of other two sides is

7.5cm.

P

X

600

Y 4.5cm Z

16. Construct a triangle ABC whose perimeter is 9cm. and the angles are in the ratio

3:4:5.

Ratio of angles=3:4:5

Sum of terms of ratio=3+4+5=12.

Measure of 1st angle=

3

12∗ 180 = 45° .

Measure of 2nd

angle= 4

12∗ 180 = 60°.

Measure of 3rd

angle= 5

12∗ 180 = 75°.

R S

A

450 45

0 60

0 60

0

P 13.5cm B C Q

17. Construct a triangle ABC whose perimeter is 12cm, and the angles are in the ratio

2:3:5.

Ratio of angles=2:3:5.

Sum of 3 terms of ratio=2+3+5=10.

Measure of 1st angle=

2

10∗ 180 = 36°.

Measure of 2nd

angle=3

10∗ 180 = 54°.

Measure of 3rd

angle=5

10∗ 180 = 90°.

R S

A

450 45

0 60

0 60

0

P 13.5cm B C Q

18. Construct a triangle ABC in which BC=4.5cm. B=350 and difference between the

other two sides is 2.8cm.

A

X

D

350

B 4.5cm C