Worksheet on Constructions of Triangles.

In this article, we are going to see solved questions and also practice questions for a better understanding of the concept of the construction of triangles.

What is the Construction of a Triangle?

A triangle is a three-sided polygon that has three edges and three vertices and the sum of the three angles (internal) of any given triangle is 180°.

Making triangles will involve using a protractor, compass, and ruler to make different angles. It has three sides, three vertices, and three angles. Construction of triangles is easy when measurements are given to us based on different properties such as SSS, SAS and ASA.

Important Formulas

To find the measure of angles or sides in a triangle:

Law of Sines: a/Sin A = b/Sin B = c/Sin C

Constructing triangles when you have two sides and the included angle:

Law of Cosines: c²= a² + b² - 2ab cos C

Pythagorean Theorem: Right angle triangle with a and b and hypotenuse c.

a² + b² = c²

Area of a Triangle: if b and h are the base and height of a triangle. 'A' is given by;

A = ½ bh

Heron's Formula: A triangle with sides a, b, c and semi-perimeter s = a + b + c / 2

A = √(s (s-a) (s-b) (s-c) )

Median of a Triangle: m = ½√(2a² + 2b² - c²)

Altitude of Triangle: A = ½bh

Centroid of a Triangle: The Centroid of a triangle is the point of concurrency of its medians.

G (x₁+ x₂+ x₃/ 3 , y₁+ y₂+ y₃/ 3)

Construction of Triangle Solved Questions

Question 1: Construct △XYZ, angle X = 50^o, angle Y = 70^o, and YZ = 7cm.

Solution:

Triangle XYZ

Draw a line segment YZ = 7cm

At point Y, use a protractor to draw an angle Y = 70^o

At point Z, use a protractor to draw an angle (where the line Y and Z meet) and mark the joining point as X = 50^o

Measure 7cm along the ray extending from X (towards the interior of angle X) to point Z.

Connect point X and Z.

△XYZ is constructed.

Question 2: Construct a triangle ABC, where AB = 6cm, BC = 5cm, and angle B=60^o

Solution:

Triangle ABC
Draw a line segment AB = 6cm.

At point B, use a protractor to draw angle B = 60^o

Using a ruler, mark the point 5cm along the ray extending from B (towards the interior of angle B) to point C.

Connect points A and C.

The △XYZ is constructed.

Question 3: Draw an isosceles triangle ABC with two sides of the triangle equal to 6 units and one side equal to 5 units.

Solution:

Triangle ABC
Using a ruler and a pencil draw a line segment AB = 5cm.

Put the compass at B and draw an arc with a length of 6cm above the line C.

Now, put the compass at A and draw an arc with the same measure of 6cm.

So that the arcs should intersect at point C.

Now, join the Points AB and AC to form an isosceles triangle ABC.

Question 4: Construct a triangle XYZ with XZ = 8cm. ∠X = 45^o and ∠Y = 65^o.

Solution:

Triangle XYZ
Use a ruler and draw a horizontal line of length 8cm. Mark X and Y on both sides of the line.

Put the center of the protractor on X and mark it as point Z.

Now place the center of the protractor on Y and look for 65^o in the protractor.

Join XZ and YZ

We formed an acute-angled triangle with the given angles.

Question 5: Construct a triangle LMN, given that LM = 7cm, MN = 9cm, and angle M = 45^o

Solution:

Triangle LMN
Draw a line segment LM = 7cm

At point M, draw angle M = 45^o

Measure 9cm along the ray extending from N (towards the interior of angle M) to point N.

Connect points L and N.

△LMN is constructed.

Question 6: Construct a triangle XYZ, XY = 10cm, YZ = 8cm, and angle X = 30^o

Solution:

Triangle XYZ
Draw a line segment XY = 10cm.

At point X, draw angle X = 30^o

Use the Law of Sines to find the length of side XZ:

XZ/sin X = XY/sin Y

XZ/ sin 30^o = 10/sin 60^o

XZ = 10×sin 30^o/sin 60^o

XZ = 5√3cm

Measure approximately 5√3cm along the ray extending from X (towards the interior of angle X) to point Z.

Connect points Y and Z.

△XYZ is constructed.

Question 7: Construct a triangle UVW, where VW = 12cm, UW = 15cm, and ∠U = 50^o

Solution:

Triangle UVW
Draw a line segment VW = 12cm

At Point V, draw angle V = 50^o

UV^{2 =}UW² + VW² - 2(UW)(VW) cos U

UV2²= 15² + 12² - 2(15)(12) cos 50^o

UV \sim9.83cm

Question 8: Construct a triangle KLM, KL = 10cm. LM = 6cm and angle L = 75^o

Solution:

Triangle KLM
Draw a line segment KL = 10cm

At point K, draw angle L = 75^o

By using the Law of Cosines to find the length of side KM:

KM²= KL² + LM² - 2(KL)(LM) cos L

KM² = 10² + 6² - 2(10)(6) cos 75^o

KM \sim 8.48cm

Question 9: Construct a triangle whose two angle measurements are 40^o and 70^o and the side length between them is 8cm.

Solution:

Triangle ACB
Draw the line of length 8cm using a ruler AC = 8cm

Put the center of the protractor on point A and measure 40^o.

Now, put the construction mark at 40^o

Using the ruler, draw a long line from A through the construction mark.

Again, place the center of the protractor on point B and measure 70^o

Now, put a mark on 70^o and more the intersection point as C.

Now, draw a line by joining points B and C.

Hence, △ABC is constructed.

Question 10: Construct a triangle ABC whose side lengths are 3cm, 5cm, and 6cm.

Solution:

Triangle ABC
Draw the longest side using ruler AB = 6cm.

Take a compass, and draw an arc above line AB from point A, whose measurement is 5cm.

Similarly from point B, draw an arc whose measurement is 3cm.

Mark the intersection point as C and join CA and CB using a ruler.

Hence, △ABC is constructed.

Practice Problems

1. Calculate the semi-perimeter of a triangle with sides of lengths 6cm, 8cm and 10cm.

2. Construct a triangle PQR with PQ = 6cm, QR = 8cm, and ∠P = 45^o.

3. Determine the length of the third side of a right triangle with sides of lengths 6cm and 8cm.(Pythagorean Theorem)

4. Construct a triangle LMN with LM = 9cm, MN = 12cm, and ∠M = 30^o.

5. Find the length of the hypotenuse of a right triangle with legs of lengths 9cm and 12cm.

6. Construct an equilateral triangle in which AB=BC=CA = 6cm. What is the measure of each angle?

7. Construct a triangle RST with RS = 8cm, ST = 10cm, and ∠S = 40^o.

8. Construct a triangle ABC in which AB = AC = 7.2cm, BC = 9cm.

9. Construct a triangle OPQ with OQ = 12cm, PQ = 9cm, and ∠O = 55^o.

10. Construct a triangle ABC with AB = 4cm, BC = 5cm, and ∠B = 90^o.

Worksheet on Constructions of Triangles.

What is the Construction of a Triangle?

Important Formulas

Construction of Triangle Solved Questions

Question 1: Construct △XYZ, angle X = 50o, angle Y = 70o, and YZ = 7cm.

Question 2: Construct a triangle ABC, where AB = 6cm, BC = 5cm, and angle B=60o

Question 3: Draw an isosceles triangle ABC with two sides of the triangle equal to 6 units and one side equal to 5 units.

Question 4: Construct a triangle XYZ with XZ = 8cm. ∠X = 45o and ∠Y = 65o.

Question 5: Construct a triangle LMN, given that LM = 7cm, MN = 9cm, and angle M = 45o

Question 6: Construct a triangle XYZ, XY = 10cm, YZ = 8cm, and angle X = 30o

Question 7: Construct a triangle UVW, where VW = 12cm, UW = 15cm, and ∠U = 50o

Question 8: Construct a triangle KLM, KL = 10cm. LM = 6cm and angle L = 75o

Question 9: Construct a triangle whose two angle measurements are 40o and 70o and the side length between them is 8cm.

Question 10: Construct a triangle ABC whose side lengths are 3cm, 5cm, and 6cm.

Practice Problems

Explore

Question 1: Construct △XYZ, angle X = 50^o, angle Y = 70^o, and YZ = 7cm.

Question 2: Construct a triangle ABC, where AB = 6cm, BC = 5cm, and angle B=60^o

Question 4: Construct a triangle XYZ with XZ = 8cm. ∠X = 45^o and ∠Y = 65^o.

Question 5: Construct a triangle LMN, given that LM = 7cm, MN = 9cm, and angle M = 45^o

Question 6: Construct a triangle XYZ, XY = 10cm, YZ = 8cm, and angle X = 30^o

Question 7: Construct a triangle UVW, where VW = 12cm, UW = 15cm, and ∠U = 50^o

Question 8: Construct a triangle KLM, KL = 10cm. LM = 6cm and angle L = 75^o

Question 9: Construct a triangle whose two angle measurements are 40^o and 70^o and the side length between them is 8cm.