Page 228 - Ganit Kaushal

Sol. Follow the steps given below:
Step 1: Draw a line PQ = 6 cm with the help of a ruler as shown in the figure (i). P 6 cm Q
(i)
Step 2: Draw a perpendicular to the
line PQ at the point P with the help of
a protractor as shown in figure (ii).
Similarly, draw a perpendicular to the
line PQ at the point Q with the help of
a protractor as shown in the adjoining P 6 cm Q P 6 cm Q
figure (iii). (ii) (iii)
After completing step 2, you will get the adjoining figure (iv). Now we have to find two points, R and
S, to complete the rectangle PQRS. R should be on the perpendicular at Q, and S should be on the
perpendicular at P.

This can be easily done with the S S R
help of a compass. Use a ruler to set
the distance from the pointer of the 4 cm 4 cm 4 cm
compass to the pencil’s lead at 4 cm.
Step 3: Without changing the setting P 6 cm Q P 6 cm Q P 6 cm Q
of the compass, place the pointer at P (iv) (v) (vi)
and draw an arc to cut the perpendicular at P. Mark this point as ‘S’ as shown in the figure (v).

Similarly, place the pointer at Q and draw an arc to cut the perpendicular at Q. Mark this point as ‘R’
as shown in the figure (vi).
Step 4: Join SR with the help of a ruler and pencil to get the desired rectangle PQRS.

Now, it is clear that PS = QR, and ∠P = ∠Q = 90° (by construction). S R
How long is side SR, and what are the measures of ∠R and ∠S? Is SR = PQ =
6 cm and ∠R = ∠S = 90° too? 4 cm 4 cm
Yes, they are. 90° 90°
Therefore, PQRS is a rectangle as it satisfies both the rules R1 and R2. P 6 cm Q

Exploring Diagonals of Rectangles and Squares

Construct a rectangle PQRS of any measurement of your choice. Join PR and QS. P d e Q
c f
These two lines are called the diagonals of the rectangle.
Observe that a diagonal divides each pair of opposite angles into two smaller angles.
In the figure, the diagonal PR divides ∠R into two smaller angles, ∠g and ∠h. It b g
also divides ∠P into ∠c and ∠d. a h
S R
Likewise, the diagonal QS divides ∠Q into two smaller angles ∠e and ∠f and it
also divides ∠S into ∠a and ∠b.
Now, based on it, can you answer the following questions?
Compare the lengths of the diagonals. Are they equal?
Yes, they are. Great answer! Absolutely correct.

Do the diagonals divide opposite angles into two equal smaller parts? That is, ∠a = ∠b, ∠e = ∠f,
∠c = ∠d, and ∠g = ∠h?
No, they are not equal. Great! Correct again.

226 Mathematics-6