Playing with Constructions | IT

Question 11

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.

Solution

The distance between X and Y changes as they move. We need to find the smallest and largest possible distances.

Step 1 — Understanding the Rectangle Let us draw a rectangle ABCD. The length AB is 7 cm. The width BC is 4 cm. Point X is on side AD. Point Y is on side BC.

Diagram 1

Step 2 — Finding the Closest Distance Let us think about the shortest path. This path is between X and Y. Sides AD and BC are parallel. The shortest distance between parallel lines is perpendicular. If X and Y are at the same height. The line segment XY is perpendicular. It is perpendicular to AD and BC. So the distance XY equals the rectangle's length. The length of the rectangle is AB.

$XY = AB$ $XY = \mathbf{7 \text{ cm}}$

So, X and Y are closest when they are directly opposite. This means $AX = BY$ . Or, we can say $DX = CY$ . The distance between them will be 7 cm.

Diagram 2

Step 3 — Finding the Farthest Distance Let us think about the longest path. This path is between X and Y. Imagine X is at one corner of side AD. Let X be at point A. Then Y can be anywhere on side BC. The distance AY will be longest. This happens when Y is at corner C. Corner C is farthest from A on BC. So, the distance AC is a diagonal. It is a diagonal of the rectangle. Let us use the Pythagorean theorem. This finds the length of AC. Triangle ABC is a right-angled triangle. AB is 7 cm. BC is 4 cm.

$AC^2 = AB^2 + BC^2$ $AC^2 = \mathbf{7}^2 + \mathbf{4}^2$ $AC^2 = \mathbf{49} + \mathbf{16}$ $AC^2 = \mathbf{65}$ $AC = \sqrt{\mathbf{65}}$

The value of $\sqrt{65}$ is about 8.06 cm. Similarly, if X is at D. The farthest point Y on BC is B. The distance DB is also a diagonal. So, X and Y are farthest at opposite corners. This means X is at A and Y is at C. Or X is at D and Y is at B. The distance between them will be $\sqrt{\mathbf{65} \text{ cm}}$ .

Diagram 3

Step 4 — Intuition Our intuition tells us something. The closest points are directly opposite each other. The line connecting them is perpendicular. It is perpendicular to both lines. For the farthest points, we want the longest segment. This happens at the 'extreme' ends. These extreme ends are the corners. Connecting opposite corners gives the longest distance.

Answer

(i) X and Y will be at their closest when they are at the same height. This means $AX = BY$ . The distance will be 7 cm. (ii) X and Y will be at their farthest when they are at opposite corners of the rectangle. This means X is at A and Y is at C, or X is at D and Y is at B. The distance will be $\sqrt{\mathbf{65} \text{ cm}}$ . (iii) My intuition says the closest points are directly opposite each other, and the farthest points are at the corners that are furthest apart.

More questions in IT

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Q10

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

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

Q11

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.

Q12

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?

Q13

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:

Q14

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:

Q15

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

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

Q16

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

Q17

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?

Q18

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?

Q19

Explore: Can the rectangle now be completed?

Q20

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

Q21

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

two identical squares;
three identical squares.

Q22

Construct

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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?

Q23

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.

Q24

Shadings

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

Q25

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.

Q26

5. Square with more Holes

Q27

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!

Q28

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.

Q29

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.

Q30

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

Q31

Construct

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

Q32

Construct

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?

Q33

Construct

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

Q34

Construct

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

Q35

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

Q36

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?

Q37

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

Q38

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

Q39

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?

Q40

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?