Playing with Constructions

Observe the following figures and try drawing them freehand.

Mark a point ‘P’ in your notebook. Then, mark as many points as possible, in different directions, that are 4 cm away from P.

Think: Imagine marking all the points of 4 cm distance from the point P. How would they look?

Try to draw it and verify if it is correct by taking some points on the curve and checking if their distances from P are indeed 4 cm. Explore, if you have not already done so, and see if a compass can be used for this purpose.

You can start by marking a few points of distance 4 cm from P using the compass. How can this be done?

Having explored the use of a compass, go ahead and recreate the images in Fig. 8.1. Can you make the figures look as good as the figures shown there? Try again if you want to!

Also, has the use of instruments made the construction easier?

Construct

1. A Person

How will you draw this?

2. Wavy Wave

Construct this.

Make other artwork of your choice with a ruler and a compass.

Explore

How should the rectangle be constructed so that the diagonal divides the opposite angles into equal parts?

How will you record your observations? First, identify the parameters that need to be tracked. They are the sides of the rectangle and the 8 angles formed by the two diagonals. Are there any other measurements that you would want to keep track of?

In your experimentation, did you consider the case when all four sides of the rectangle are equal? That is, did you consider the case of a square? See what happens in this special case!

Math Talk

What general laws did you observe with respect to the angles and sides? Try to frame and discuss them with your classmates.

How can one be sure if the laws that you have observed will always be true?

What radius should be taken in the compass to get this half circle? What should be the length of AX?

Take a central line of a different length and try to draw the wave on it.

Try to recreate the figure where the waves are smaller than a half circle (as appearing in the neck of the figure, 'A Person'). The challenge here is to get both the waves to be identical. This may be tricky!

Draw the rectangle and four squares configuration (shown in Fig. 8.3) on a dot paper.

What did you do to recreate this figure so that the four squares are placed symmetrically around the rectangle? Discuss with your classmates.

Identify if there are any squares in this collection. Use measurements if needed.

Think: Is it possible to reason out if the sides are equal or not, and if the angles are right or not without using any measuring instruments in the above figure? Can we do this by only looking at the position of corners in the dot grid?

Draw at least 3 rotated squares and rectangles on a dot grid. Draw them such that their corners are on the dots. Verify if the squares and rectangles that you have drawn satisfy their respective properties.

Observe the way a compass is made. What can one draw with the compass? Explore!

Take a point on the circle. What will be its distance from P—equal to 4 cm, less than 4 cm or greater than 4 cm? Similarly, what will be the distance between P and another point on the circle?

3. Eyes

How do you draw these eyes with a compass?

What shapes are these? Yes, these are our familiar squares and rectangles. But what makes them squares and rectangles?

Which of the following is not a name for this square?

1. PQSR
2. SPQR
3. RSPQ
4. QRSP

Can you see why PS should be 6 cm long?

How long is the side RS and what are the measures of $\angle R$ and $\angle S$?

Draw a rectangle with sides of length 4 cm and 6 cm. After drawing, check if it satisfies both the rectangle properties.

Draw a rectangle of sides 2 cm and 10 cm. After drawing, check if it satisfies both the rectangle properties.

Is it possible to construct a 4-sided figure in which—

• all the angles are equal to 90° but
• opposite sides are not equal?

In a rectangle ABCD with AB = 7 cm and BC = 4 cm, X is a point that can be moved anywhere along the side AD, and Y is a point that can be moved anywhere along the side BC.

Q. At which positions will the points X and Y be at their closest? When do you think they will be the farthest? What does your intuition say? Discuss with your classmates.

In a rectangle ABCD with AB = 7 cm and BC = 4 cm, X is a point that can be moved anywhere along the side AD, and Y is a point that can be moved anywhere along the side BC.

Q. Verify your guesses by placing the points X and Y on the sides and measure how near or far they are. The distance between X and Y can be obtained by measuring the length of the line XY.

(i) How does the minimum distance between the points X and Y compare to the length of AB? (ii) Change the positions of X and Y to check if there are other positions where they are at their nearest or farthest. How will you keep track of the lengths XY for different positions of X and Y?

In a rectangle ABCD with AB = 7 cm and BC = 4 cm, X is a point that can be moved anywhere along the side AD, and Y is a point that can be moved anywhere along the side BC.

Q. Suppose here are some of the positions of X and Y that you have considered. Find the length of XY for each case shown in the list below:

Q. Is there a shorthand way of writing it down? In all the sentences, only the position of X, Y and the length XY changes. So we could write this as shown in the table below:

Have you checked what happens to the length XY when X and Y are placed at the same distance away from A and B, respectively? For example, as in the cases like these:

and so on.

In each of these cases, observe

1. how the length XY compares to that of AB and
2. the shape of the 4-sided figure ABYX.

How does the farthest distance between X and Y compare with the length of AC? BD?

Explore

What about constructing a rectangle that can be divided into two identical squares? Can you try it?

It is wise to first plan and then construct. But how do we plan? Can you think of a way?

To draw the rectangle $ACDF$, one could assign any length to $AF$. For example, if we assign $AF = 4\text{ cm}$, then what must the length of $AC$ be?

Explore: Can the rectangle now be completed?

With this idea, try constructing a rectangle that can be divided into three identical squares.

Give the lengths of the sides of a rectangle that cannot be divided into—

• two identical squares;
• three identical squares.

Construct

1. A Square within a Rectangle

Construct a rectangle of sides 8 cm and 4 cm. How will you construct a square inside, as shown in the figure, such that the centre of the square is the same as the centre of the rectangle?

Falling Squares

Construct the figure shown below, where each is a square of side 4 cm. Make sure that the squares are aligned the way they are shown.

Now, try this: Construct the figure with squares of side 3 cm, 5 cm, and 7 cm as shown.

Shadings

Construct this. Choose measurements of your choice. Note that the larger 4-sided figure is a square and so are the smaller ones.

4. Square with a Hole

Observe that the circular hole is the same as the centre of the square.

Hint: Think where the centre of the circle should be.

5. Square with more Holes

6. Square with Curves

This is a square with 8 cm sidelengths.

Hint: Think where the tip of the compass can be placed to get all the 4 arcs to bulge uniformly from each of the sides. Try it out!

8.5 Exploring Diagonals of Rectangles and Squares

Consider a rectangle PQRS. Join PR and QS. These two lines are called the diagonals of the rectangle.

Compare the lengths of the diagonals. First predict the answer. Then construct a rectangle marking the points as shown and measure the diagonals.

Observe that a diagonal divides each of the pair of opposite angles into two smaller angles. In the figure, the diagonal PR divides angle R into two smaller angles which we simply call g and h. The diagonal also divides angle P into c and d. Are g and h equal? Are c and d equal?

First predict the answers, and then measure the angles. What do you observe? Identify pairs of angles that are equal.

Check if ABCD is indeed a rectangle satisfying properties R1 and R2.

Construct

1. Construct a rectangle in which one of the diagonals divides the opposite angles into $50^\circ$ and $40^\circ$.

Construct

2. Construct a rectangle in which one of the diagonals divides the opposite angles into $45^\circ$ and $45^\circ$. What do you observe about the sides?

Construct

3. Construct a rectangle one of whose sides is 4 cm and the diagonal is of length 8 cm.

Construct

4. Construct a rectangle one of whose sides is 3 cm and the diagonal is of length 7 cm.

Was it necessary to draw two full circles to get the point A? We only needed part of both the circles.

Having obtained point A, what remains is the construction of the remaining arc. How do we do it?

Can we use the fact that A is of distance 5 cm from both B and C?

Construct a bigger house in which all the sides are of length 7 cm.

Try to recreate 'A Person', 'Wavy Wave', and 'Eyes' from the section 'Artwork', using ideas involved in the 'House' construction.

Is there a 4-sided figure in which all the sides are equal in length but is not a square? If such a figure exists, can you construct it?

B) (From Construct above (page no. 211).)

For the purpose of construction, let us take the side lengths to be of 5 cm. Consider this figure.

We need to identify only one more point to make this a 4-sided figure. That point, let us call it D, should be 5 cm from both B and C. How can such a point be found? Can any of the ideas used in the 'House' problem be used here?