Constructing a Triangle When its Sides are Given

Class 7 Maths · Chapter 7: A Tale of Three Intersecting Lines · NCERT Ganita Prakash section 7.2

Learn

To make a triangle with sides 4, 5 and 6 cm: draw base AB = 4 cm. Draw an arc of 5 cm from A and an arc of 6 cm from B. They cross at C, and △ABC has AC = 5 cm and BC = 6 cm. Full circles are not needed, only arcs long enough to cross.

AB = 4 · AC = 5 · BC = 6

Not every set works. Try 3, 4 and 8 cm: the arcs never meet. Why? The direct path between two points is always shorter than going round through a third point. In a triangle with sides 10, 15, 30, the direct path 30 would be longer than 10 + 15 = 25: impossible!

30 > 10 + 15 → no triangle

Triangle inequality: three lengths make a triangle only if each is less than the sum of the other two. Shortcut: check only the longest. For 4, 5, 8: 8 < 4 + 5 = 9, so it works. For 2, 3, 6: 6 > 2 + 3, so it fails.

longest < sum of the other two

Picture the circles around A and B (base = the longest side, radii = the two shorter). If the radii add to more than AB, the circles cross: a triangle. If they add to exactly AB, the circles just touch: flat, no triangle. If less, they do not meet. So with sides 1 and 100, the third side must be strictly between 99 and 101.

1, 100: 99 < third < 101

Explore: triangle or not?

1. Sort the sets of side lengths.

Remember

• Given three sides: base, then two arcs from its ends; they cross at the third vertex.
• Triangle inequality: the longest side must be less than the sum of the other two.

Quick check

Which set of lengths can make a triangle?

1. 5, 10, 12
2. 1, 1, 5
3. 3, 6, 9
4. 10, 20, 35

Think about it

Two sides of a triangle are 5 cm and 5 cm. What can the third side be?

Practice

Practise with fresh questions at three levels (easy, medium, hard), some answered hands-on, with hints and worked answers. Get 4 of your last 5 right to finish the topic.