NCERT Solutions Ganita Prakash (Part 1) Chapter 7 Are Triangles Possible for any Lengths? — In-text Questions
Book page 151 Updated on2026-09-05
Q1.
Construct a triangle with sidelengths 3 cm, 4 cm, and 8 cm. What is happening? Are you able to construct the triangle?
Answer
No. The two arcs never meet, so the third vertex simply does not exist.
Draw the base AB = 8 cm Arc from A of radius 3 cm reaches only 3 cm along Arc from B of radius 4 cm reaches only 4 cm back 3 + 4 = 7 cm, but the gap to be covered is 8 cm 7 < 8 → the arcs fall 1 cm short of each other
With a base of 8 cm, arcs of 3 cm and 4 cm cannot reach each other.
Why it happens: The two shorter sides together have to bridge the base. Here they add up to only 7 cm while the base is 8 cm, so however you tilt them they can never join. No triangle exists with sides 3 cm, 4 cm and 8 cm.
Q2.
Here is another set of lengths: 2 cm, 3 cm, and 6 cm. Check if a triangle is possible for these sidelengths.
Answer
Not possible. The same thing happens, only worse.
Longest length = 6 cm Sum of the other two = 2 + 3 = 5 cm 5 < 6 → the arcs fall short by 1 cm No triangle exists
Check it yourself: Draw AB = 6 cm, then an arc of 2 cm from A and an arc of 3 cm from B. A clear white gap is left between them.
Q3.
Try to find more sets of lengths for which a triangle construction is impossible. See if you can find any pattern in them.
Answer
Here are five impossible sets, and the pattern behind all of them.
Set of lengths
Longest
Sum of other two
Triangle?
1, 2, 5
5
3
No
2, 4, 9
9
6
No
3, 5, 8
8
8
No (arcs only touch)
6, 7, 20
20
13
No
1, 1, 3
3
2
No
The pattern: a triangle is impossible exactly when
longest length ≥ sum of the other two lengths
Why it happens: The two smaller sides have to stretch from the two ends of the longest side and meet somewhere above it. Their total length must be more than the base — if it is exactly equal they flatten onto the base and give a straight line, and if it is less they cannot meet at all.
Try This: Make an impossible set of your own by picking any two lengths and making the third one bigger than their sum, for example 7, 9 and 20.