NCERT Class 7 Ganita Prakash 2 Chapter 01 Geometric Twins PDF Download

Chapter 1: Geometric Twins

1.1 Geometric Twins

The symbol on this signboard needs to be recreated on another board.

[Figure: A checkmark symbol on a signboard, See in your textbook]

How do we do it?

One way is to trace the outline of this symbol on tracing paper to reconstruct the figure. But this is difficult for big symbols. What else can we do?

Can we take some measurements that would allow us to exactly recreate this figure? If yes, what measurements should we take?

Let us name the corner points of this symbol as shown.

[Figure: A simple angle with corner points labeled A, B, and C, See in your textbook]

Are the arm lengths AB and BC sufficient to exactly recreate this figure?

Suppose these lengths are AB = 4 cm, BC = 8 cm. We observe that several such symbols can be constructed with the same lengths.

[Figure: Four different angles with the same arm lengths but different angles between them, labeled with points A, B, and C, See in your textbook]

To get the exact replica, would it help to take any other measurement?

The measure of \( \angle ABC \), along with the two arm lengths AB and BC, fix the shape and size of this figure.

Can you draw the symbol if it is known that AB = 4 cm, BC = 8 cm, and \( \angle ABC = 80° \)?

These three measurements can help us create an exact replica of the symbol on the signboard. Figures that are exact copies of each other or have the same shape and size are said to be congruent. Congruent figures can be superimposed exactly, one over the other.

Teacher's Note

When you see the word "congruent," remember it means the figures are identical in shape and size. You can flip or rotate one figure before placing it on top of another - if they match perfectly with no gaps or overlaps, they are congruent. Just matching one side or one angle is not enough; you need to check that all corresponding parts are equal.

Two congruent figures are shown below. You could use a tracing paper to trace the first figure and superimpose it on the second one. You will find that they fit exactly, one over the other.

[Figure: Two congruent irregular quadrilaterals, one pink and one beige, See in your textbook]

Note that while checking for congruence, a figure can be rotated or flipped before superimposing it on the other figure. So, the following pairs of figures are also congruent to each other.

[Figure: Two pairs of congruent shapes - one pair showing two semicircles, See in your textbook]

Let us get back to the symbol we saw on the signboard. Suppose there are two such symbols that look identical and we need to confirm that they are indeed congruent. Can we use their measurements to verify this?

If it is known that both symbols have the same arm lengths, can it be concluded that the two symbols are congruent?

We have seen that there can exist several such non-congruent figures with different angles between the given arm lengths. Fixing the angle determines the shape and size of the figure.

Thus, if both symbols have the same arm lengths and angle, we can be sure that the figures are congruent.

Teacher's Note

In problems asking you to verify congruence, do not assume two figures are congruent just because one or two measurements match. Always check all the measurements mentioned in the problem. For the angle and two arm lengths, you have enough information to guarantee congruence - but only if all three pieces of information are the same in both figures.

Figure it Out

1. Check if the two figures are congruent.

   [Figure: Two angles of different sizes, See in your textbook]

2. Circle the pairs that appear congruent.

   [Figure: Four pairs of shapes - a teardrop and its mirror image, a cloud shape pair, a star shape pair, and a leaf-like shape pair, See in your textbook]

3. What measurements would you take to create a figure congruent to a given:
   1. Circle
   2. Rectangle

Key Points

• Congruent figures are exact copies of each other - they have the same shape and size and can be superimposed perfectly, even if one is rotated or flipped.
• For a figure like an angle with two arms, you need three pieces of information to create an exact copy: the lengths of both arms and the angle between them.
• Just knowing some measurements is not always enough; knowing only the arm lengths allows many different figures to be drawn, so you must also know the angle.

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