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NCERT Solutions for Class 5th Maths Chapter 7 What meets at a marked point; squares and triangles together — Tiling and Tessellation

Book page 96 Updated on2026-09-19

Q1.
Look at the pattern given below. What shapes are coming together at the marked points? Are the same set of shapes coming together at these points? Continue the pattern and colour it appropriately.
The dotted grid on page 96, with the four marked points. In the printed book the pattern itself is missing from this grid — only the marked points are shown.
Answer

First, a note about the book. In the printed page the top dotted box is empty — only the four red dots are shown, and the tiles themselves have not been printed. So there is nothing there to read off. Use the second pattern on the same page (the one with squares and triangles) for this question, and use the method below on any tiling.

How to answer “what is coming together at a point”

  1. Put your pencil tip on the marked point.
  2. Go round it once, like the hand of a clock. Name each tile you pass over.
  3. Write the list in order — for example: square, triangle, square, triangle, triangle.
  4. Do the same at the next marked point and compare the two lists.
  5. If both lists have the same shapes in the same order, the answer is “yes, the same set of shapes comes together at every point”.
Check with the corner sizes:
square corner = 90  ·  equilateral-triangle corner = 60
3 triangles + 2 squares = 60 + 60 + 60 + 90 + 90 = 360 ✓

Sample answer: At each marked point three equilateral triangles and two squares come together. The same five shapes meet at every marked point, so the pattern is the same all over. To continue it, keep placing squares and triangles so that this same group of five appears at every new corner, then colour all the squares one colour and all the triangles another.

If your copy of the book does show a pattern in that box, use the same five steps on it: walk round each red dot, list the shapes, and compare.
Q2.
Here is a tiling pattern made using two different shapes — squares and triangles. Are the triangles equilateral? Why or why not?
The tiling pattern on page 96, with the marked points in red.
Answer

Yes, the triangles are equilateral — all three of their sides are equal.

Here is the reason, in three short steps.

  1. Look at any triangle in the pattern. Each of its sides is also the side of a square that touches it.
  2. All four sides of a square are equal. So every side of the triangle is the same length as a side of a square.
  3. All the squares in the pattern are the same size. So all three sides of the triangle are equal — that is exactly what “equilateral” means.
At the big red dot: 2 squares + 3triangles The dotted triangle is the one that fills thelast gap.
Every side of every orange triangle is also the side of a purple square, so all the triangle's sides are equal.
Why the pattern works: At the big red dot, two squares and three triangles meet. 90 + 90 + 60 + 60 + 60 = 360, one full turn. Because the corners add up exactly, the tiles cover the floor with no gap and no overlap.
Check it yourself: Measure any triangle side and any square side in the picture with your ruler. They come out the same. That is the proof, done with a ruler instead of words.
Q3.
What shapes are coming together at the marked points? Are the same set of shapes coming together at these points? Continue the pattern and colour it appropriately.
The tiling pattern on page 96, with the marked points in red.
Answer

Three equilateral triangles and two squares come together at each marked point. Yes — the same five shapes meet at every marked point.

Walk round the big red dot in the picture above and count what you pass:

Going round the pointShapeCorner size
1stSquare90
2ndTriangle60
3rdSquare90
4thTriangle60
5thTriangle60
Total2 squares + 3 triangles360

How to continue the pattern

  1. Find a corner that is not yet full. Count the shapes already round it.
  2. Work out what is missing. Each corner needs 2 squares and 3 triangles in all.
  3. Draw the missing tile with its side exactly along a side that is already there.
  4. Move to the next unfinished corner and do the same.
  5. Colour last: all squares one colour, all triangles another.
Why the same set appears everywhere: The tiling was built by repeating one small group of tiles over and over. Repeating a group means every point ends up looking like every other point — that is what makes it a pattern and not just a jumble.
Tip: Work outwards from the middle, one ring at a time. If you jump about, you will leave holes that are hard to fill later.
Q4.
Create similar patterns using other cutouts of shapes.
Answer

Use the cut-outs from the sheet at the end of your book — squares, equilateral triangles, regular hexagons and regular pentagons.

Steps

  1. Choose your shapes. Start with two kinds, for example hexagons and triangles.
  2. Put one shape down in the middle of a sheet of paper.
  3. Add shapes side by side, always matching a whole side to a whole side. Never let one cut-out climb on another.
  4. Check every corner. The corners meeting there must add up to 360 — no gap, no overlap.
  5. Trace round the cut-outs with a pencil, lift them off, and colour your pattern.

Sample answer — three patterns that work

PatternShapes usedWhat meets at each pointCheck
1Squares only4 squares90 × 4 = 360
2Triangles only6 triangles60 × 6 = 360
3Hexagons and triangles2 hexagons + 2 triangles120 + 120 + 60 + 60 = 360
4Squares and triangles2 squares + 3 triangles90 + 90 + 60 + 60 + 60 = 360
One that will not work: Pentagons on their own. Try them and watch the thin gap appear every time. Finding out what does not work is just as useful as finding what does.
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