Page 229 - Ganit Kaushal

Then, by measuring all the angles, can you identify pairs of angles that are equal?

Yes, we got that ∠a = ∠e, ∠b = ∠f, ∠c = ∠g, and ∠d = ∠h. Great answer!
This is always true. Such a pair of equal angles are called alternate angles.
How should a rectangle be constructed so that a diagonal divides the opposite angles into equal parts?

A rectangle with equal sides, i.e., a square — has diagonals that divide the opposite angles into equal
parts. Fantastic! You hit the bullseye.

Some More Exploration in Rectangles

Consider a rectangle ABCD with AB = 7 cm and BC = 4 cm. A 7 cm B
1 cm
Imagine a point X that can be moved anywhere along the side AD. Similarly, X 2 cm
imagine a point Y that can be moved anywhere along the side BC. Note that X can 4 cm Y
also be placed on the end points A or D. Similarly, Y can also be placed on the end D C
points B or C. (see adjoining figure (i)) (i)

Now, based on it, can you answer the following questions?
At which positions will the points X and Y be closest to each other?
You are absolutely right! X and Y will be closest when they are at A and B, or A = X B = Y
at D and C, or at equal distances from A and B, respectively. It can be easily X Y
observed from the adjoining figure (ii). X Y

How does this minimum distance between the points X and Y compare to the D C
length of AB? (ii)
Obviously, this minimum distance will be equal to AB.
You are absolutely right again! In fact, this minimum distance will also be equal to DC.
[ In a rectangle, opposite sides are equal. (see figure (ii))]

So, we conclude that:
Minimum distance between X and Y = AB = DC.
When do you think the points X and Y will be farthest apart?

Points X and Y will be farthest apart when they are at opposite corners that A B = Y
is, when X is at A and Y is at C, or X is at D and Y is at B. Great! You are
absolutely right again. In fact, in that case, the distance between X and Y will
be equal to the diagonals AC or BD (see figure (iii)). So, we conclude that:
D = X C
Maximum distance between X and Y = AC = BD. (iii)

2. When the Division of Opposite Angles is Given

Construction 7: Construct a rectangle in which one of the diagonals divides the opposite angles into 60°
and 30°. (NCERT)

Note
As advised earlier, start by making a rough diagram with the given measurements marked on it. A rough
diagram always helps in the construction and in deciding in what order the parts should be drawn.

Playing with Constructions 227