8. Playing with Constructions

They form a circle with centre P and radius 4 cm.

Every point on the circle is exactly 4 cm from P.

1. Open the compass against a ruler so that the pointed tip and the pencil are 4 cm apart.
2. Keep the pointed tip fixed on P.
3. Turn the compass all the way round, moving only the pencil.

AB = 8 cm has two equal waves, so AX = 8 ÷ 2 = 4 cm.

The half circle stands on AX, so its radius = 4 ÷ 2 = 2 cm.

Put the tip of the compass at the middle of AX (2 cm from A) and draw a half circle above the line. Draw the second half circle on XB below the line.

Example: Take AB = 10 cm. Then AX = 5 cm and the radius = 2.5 cm. Draw one half circle above AX and one below XB.

Keep a radius bigger than half of AX. Put the compass tip below the line for the upper wave and above the line for the lower wave, at the same distance from the line. Then both waves will be identical.

PQSR is not a valid name. In PQSR, we jump from Q to S across the square. The corners must be named in order around the square.

Draw the rectangle first. Then draw the four squares at its four corners, leaving the same number of dots (one dot diagonally) between each square and the rectangle corner.

Only A is a square.

In A, each side goes 4 dots across and 4 dots up or down. So all sides are equal and all corners are right angles.

B and C have equal sides, but their corners are not right angles (one diagonal is longer than the other). D has right angles, but its sides are not all equal, so it is a rectangle.

Think: Yes. By counting the dots moved across and up for each side, we can check equal sides and right angles without measuring.

Example square: start at a dot, go 2 dots right and 1 dot up, then 1 dot left and 2 dots up, then 2 dots left and 1 dot down, and come back.

Example rectangle: sides of “2 right, 1 up” and “2 left, 4 up”.

Check with a ruler and a protractor that the sides and the angles satisfy the properties.

RS = 6 cm. ∠R = ∠S = 90°.

1. Draw AB = 6 cm.
2. At A and at B, draw perpendiculars to AB.
3. Mark D on the perpendicular at A and C on the perpendicular at B, each 4 cm from AB (use a compass).
4. Join DC.

Check: AB = DC = 6 cm, AD = BC = 4 cm and ∠A = ∠B = ∠C = ∠D = 90°. Both properties are satisfied.

Draw PQ = 10 cm. Draw perpendiculars at P and Q. Mark S and R, 2 cm from PQ. Join SR.

Check: PQ = SR = 10 cm, PS = QR = 2 cm and all angles are 90°.

No. If all four angles are 90°, the opposite sides always turn out to be equal. Such a figure is always a rectangle.

Closest: when X and Y are at the same distance from A and B. Then XY = 7 cm, which is equal to AB.

Farthest: when X is at A and Y is at C (or X is at D and Y is at B). Then XY is a diagonal, about 8.1 cm.

1. XY is equal to AB (7 cm).

2. ABYX is a rectangle.

The farthest distance is equal to the diagonal AC (and BD). AC = BD.

Two squares: the length must be 2 times the breadth. If AF = 4 cm, then AC = 8 cm.

1. Draw AF of any length.
2. Draw a perpendicular to AF at A.
3. Open the compass to the length AF. From A, mark B on the perpendicular. From B, mark C with the same opening.
4. Draw perpendiculars at F and C, and mark E and D with the same compass opening. Join CD and BE.

Three squares: do the same, but mark the length AF three times along the perpendicular. The length becomes 3 times the breadth.

Two squares: any rectangle whose length is not 2 times its breadth, for example 5 cm × 3 cm.

Three squares: any rectangle whose length is not 3 times its breadth, for example 7 cm × 2 cm.

The square has side 4 cm (the breadth of the rectangle).

Its sides are 2 cm from each short side of the rectangle, because (8 − 4) ÷ 2 = 2 cm.

Construct the 8 cm × 4 cm rectangle. Mark points 2 cm from each end on the top and bottom sides. Join them to get the square.

Construct the first square. Each new square starts at the top-right corner of the previous one, with its bottom side on the line of the previous square’s top side. Use perpendiculars and a compass to mark equal sides.

Construct a big square (for example 8 cm). Divide it into 4 equal squares of 4 cm, and divide some of these again into 2 cm squares. Draw the diagonals and shade as shown.

The centre of the circle should be at the point where the two diagonals of the square meet.

Divide the square into 4 equal smaller squares. The centre of each hole is where the diagonals of a small square meet.

For each side, place the tip of the compass on the line through the middle of that side, at the same distance outside (or inside) the square every time. Draw an arc from one corner to the other. Using the same radius and distance for all four sides makes the arcs bulge equally.

The two diagonals of a rectangle are equal in length.

A diagonal divides a corner into two angles that add up to 90°. These two angles are not equal in a rectangle that is not a square.

The angle that a diagonal makes with a side at one end is equal to the angle it makes with the opposite side at the other end.

All four sides must be equal, that is, the rectangle must be a square. Then each angle is divided into 45° and 45°.

1. Draw AB of any length, say 6 cm.
2. Draw a perpendicular to AB at B.
3. At A, draw a ray making 40° with AB. Let it meet the perpendicular at C. (Now ∠CAB = 40° and ∠BCA = 50°.)
4. Draw a perpendicular to AB at A and a perpendicular to BC at C. They meet at D. ABCD is the rectangle.

Follow the same steps with a 45° angle at A.

We observe that all four sides become equal. The rectangle is a square.

1. Draw DC = 4 cm.
2. Draw a line l perpendicular to DC at C.
3. With D as centre and radius 8 cm, draw an arc to cut l at B.
4. Draw perpendiculars to DC at D and to BC at B. They meet at A. ABCD is the rectangle.

Same steps: draw DC = 3 cm, a perpendicular at C, and an arc of radius 7 cm from D to get B. Complete the rectangle with perpendiculars at D and B.

1. Draw the square BCED with sides 7 cm (draw DE = 7 cm, perpendiculars at D and E, mark B and C 7 cm up, join BC).
2. With B as centre and radius 7 cm, draw an arc above BC. With C as centre and the same radius, draw another arc. They meet at A.
3. Join AB and AC.
4. With A as centre and radius 7 cm, draw the arc from B to C. Draw the door as in the given figure.

For each curve, first find the point where the compass tip must go. It must be at the same distance from the two ends of the curve. Draw arcs of the same radius from both ends; their meeting point is the centre for the curve.

Yes. Such a figure is called a rhombus. Its angles are not 90°.

1. Draw AC = 5 cm.
2. Draw AB = 5 cm making any angle other than 90° with AC, for example 60°.
3. With B and C as centres and radius 5 cm, draw arcs. They meet at D.
4. Join BD and CD. ABDC has all sides 5 cm but it is not a square.