NCERT Solutions for Class 5th Maths Chapter 7 Cutting a rhombus into four triangles and building with them — Shapes from the Rhombus
Book page 96–97 Updated on2026-09-19
Q1.
You will find a copy of this rhombus at the end of the book. Cut out the triangular pieces for the following activities. What geometrical shapes can you make by fitting 2 of these triangles together? Trace the shapes you created.The rhombus on page 96, divided into four triangles.
Answer
With 2 pieces you can make five different shapes: two kinds of triangle, a kite, a parallelogram and a rectangle.
First, look at one piece. The rhombus was cut along both its long lines (the diagonals), so all four pieces are the same. Each piece is a right-angled triangle: it has one square corner, one short side, one long side and one slanting side.
The two diagonals cut the rhombus into 4 equal right-angled triangles.
Now join two pieces, always side to side.
Join them along…
How
Shape you get
Sides
the long side
as a mirror
A tall isosceles triangle
3
the short side
as a mirror
A wide isosceles triangle
3
the slanting side
as a mirror
A kite
4
the slanting side
turned half a turn
A rectangle
4
a short or long side
turned half a turn
A parallelogram
4
Five shapes from the same two pieces. The dotted line shows where they are joined.
Trace them: Lay the two pieces down, draw round the outside with a pencil, then lift them off. Write the name of the shape inside your tracing.
Q2.
How many different types of triangles can you make? Now, observe and measure the sides of these triangles. What do you notice?
Answer
You can make 2 different types of triangle. Both of them have exactly 2 equal sides — they are isosceles triangles.
What you see when you measure
Triangle 1 — the tall one. Join two pieces along their long sides. Its base is the short line of the rhombus. Its other two sides are two sides of the rhombus, so they are equal.
Triangle 2 — the wide one. Join two pieces along their short sides. Its base is the long line of the rhombus. Again the other two sides are sides of the rhombus, so they are equal.
Measure with your ruler. In both triangles the two slanting sides come out the same, and the base is different from them.
Triangle 1: side, side, short diagonal → two sides equal Triangle 2: side, side, long diagonal → two sides equal A triangle with exactly two equal sides is called an isosceles triangle.
Why the two sides are always equal: A rhombus has all four sides the same length. When you cut it in half along a diagonal, the two outside edges of each half are two sides of the rhombus — so they must be equal.
Fold and check: Cut out one of these triangles, then fold it in half so the two equal sides lie on top of each other. The two bottom corners land exactly on each other. That shows the triangle also has 2 equal angles.
Q3.
Trace the isosceles triangles on a paper and cut them out. Fold them in half. What do you notice about their angles?
Answer
The two bottom angles land exactly on top of each other. So each isosceles triangle has 2 equal angles.
Steps
Trace the triangle on plain paper and cut it out.
Find the two equal sides with your ruler.
Fold so that one equal side lies exactly along the other. Press the fold flat.
Look at the two bottom corners. They cover each other completely — no bit sticks out.
Open the paper. The fold line runs from the top corner straight down to the middle of the base.
Folding is a way of measuring. If two angles cover each other, they are equal.
Why folding proves it: When you fold, every point on one half lands on the matching point of the other half. If the two bottom corners land on each other, the two angles must be exactly the same size. This works even though we have not measured a single degree.
Remember this pair:2 equal sides → 2 equal angles. The equal angles are always the two opposite the equal sides — the two at the base.
Q4.
Is it possible to make a triangle where all three sides are equal (equilateral triangle)?
Answer
No — not with the pieces of this rhombus. Every triangle you build has exactly 2 equal sides, never 3.
Measure and see (use the printed rhombus; your cut-out will give the same story):
Triangle made
The two slanting sides
The base
All three equal?
Tall one (base = short diagonal)
side of the rhombus
short diagonal — a little longer
No
Wide one (base = long diagonal)
side of the rhombus
long diagonal — much longer
No
Why it happens: The two slanting sides are always two sides of the rhombus, so they are equal to each other. The base is always a diagonal, and in this rhombus neither diagonal is the same length as a side. So you get two equal sides and one odd one — an isosceles triangle, not an equilateral one.
That is why the book gives you a separate equilateral triangle on the cut-out sheet at the end. You need one for the next activity, and the rhombus cannot give you one.
Think about it: If a rhombus had its short diagonal exactly as long as its sides, then half of it would be equilateral. Measure your cut-out carefully and see how close it comes.
Q5.
Is it possible to make a triangle where all three sides are unequal?
Answer
No. Every triangle you can build from these pieces has 2 equal sides, so you can never get three sides that are all different.
Try all the ways of joining two pieces along a whole side:
Join along the long sides → you get the tall triangle. Two sides equal.
Join along the short sides → you get the wide triangle. Two sides equal.
Join along the slanting sides → you get a kite or a rectangle. Those are four-sided shapes, not triangles at all.
Ways of getting a triangle: 2 Both of them: isosceles (2 equal sides) Triangles with no equal sides: 0
Why it happens: Whenever two pieces join to make a triangle, one piece is the mirror of the other. A mirror always makes matching pairs, and that is what gives the two equal sides. To get a triangle with three different sides you would need pieces that are not mirror images.
New word: A triangle with no two sides equal is called a scalene triangle. You will make one in the next activity, by cutting an equilateral triangle in half.