NCERT Solutions Ganita Prakash Chapter 6 Deep Dive · Estimate and Verify — In-text Questions
Book page 134 Updated on2026-09-05
Q1.
Deep Dive: In races, usually there is a common finish line for all the runners. Here are two square running tracks with the inner track of 100 m each side and outer track of 150 m each side. The common finishing line for both runners is shown by the flags in the figure which are in the center of one of the sides of the tracks. If the total race is of 350 m, then we have to find out where the starting positions of the two runners should be on these two tracks so that they both have a common finishing line after they run for 350 m. Mark the starting points of the runner on the inner track as ‘A’ and the runner on the outer track as ‘B’.
Answer
Start at the flag and walk backwards 350 m along each track.
Inner track (side 100 m) — the flag is at the middle of a side, so it is 50 m from the corner.
50 m (half a side) + 100 + 100 + 100 = 350 m
So A is the far corner of the inner square — the corner diagonally opposite the flag's side, three full sides and one half-side away.
Outer track (side 150 m) — the flag is 75 m from the corner.
75 m (half a side) + 150 m (one full side) + 125 m = 350 m
So B lies on the opposite side of the outer square, 125 m from the corner — that is, only 25 m from the other corner.
Both runners finish at the red line in the middle of the bottom side. A on the inner track and B on the outer track are each exactly 350 m away from it.
Why the starts are different: the tracks have different perimeters (400 m and 600 m). To make the race fair, the starting points are staggered so that each runner covers exactly 350 m to the same finishing line — just as in a 200 m race on an athletics track.
Q2.
Estimate and Verify: Take a rough sheet of paper or a sheet of newspaper. Make a few random shapes by cutting the paper in different ways. Estimate the total length of the boundaries of each shape then use a scale or measuring tape to measure and verify the perimeter for each shape.
Answer
This is an activity to do with a friend. Here is how to carry it out carefully:
Cut 4 or 5 odd shapes from an old newspaper — some with straight edges, some wavy.
First estimate. Look at the shape and guess its boundary length in centimetres. Write your guess down before measuring.
Then measure. For straight edges use a scale and add the side lengths. For a curved edge, lay a thread along the boundary, mark it, straighten the thread and measure it with a scale.
Make a table: shape → my estimate → measured perimeter → difference.
Shape
Estimate
Measured perimeter
Difference
Triangle piece
20 cm
23 cm
3 cm
Wavy piece
30 cm
27 cm
3 cm
Try This: After 3 or 4 shapes your estimates will come much closer to the measured values. Estimating is a skill that improves with practice — and it tells you at once if a calculated answer is sensible.
Q3.
Akshi says that the perimeter of this triangle shape is 9 units. Toshi says it can’t be 9 units and the perimeter will be more than 9 units. What do you think?
Answer
Toshi is right. The perimeter is more than 9 units.
The triangle is drawn on a dot grid. Two of its sides run along the grid (3 units + 3 units), but the third side is a slanting (diagonal) line made of 3 diagonals of unit squares.
Perimeter = 3 straight units + 3 straight units + 3 diagonal units
= 6s + 3d units
Akshi counted every diagonal step as 1 unit, which is wrong: a diagonal of a unit square is longer than its side.
Why a diagonal is longer: in any square, the diagonal joins two opposite corners, and the straight path across is always longer than one side (it is the longest line you can draw inside the square). A unit diagonal measures about 1.4 units, so
6 × 1 + 3 × 1.4 ≈ 10.2 units
which is clearly more than 9.
Tip: measure a red line and a blue line of the figure with your scale. The blue (diagonal) line will come out longer every time.