Chapter 8 exercise answers: Playing with Constructions

Open NCERT p. 191Matches NCERT’s answer

1. The wave's two humps must look identical, so the meeting point X has to be the midpoint of AB.
2. AB = 8 cm, so AX = XB = 8 ÷ 2 = 4 cm.
3. AX is the diameter of the half circle, so the compass radius is half of AX: 4 ÷ 2 = 2 cm.

AnswerRadius = 2 cm, and AX = 4 cm.

Watch this explained “The Wavy Wave — the line decides the radius”, 4:55 into Every curve in these figures is part of a circle · हिंदी में देखें

Open NCERT p. 191One way to think about it

1. Any length works, so there are many correct drawings. Here is one, with a central line of 6 cm.
2. Draw a line PQ = 6 cm. Mark its midpoint Y, so PY = YQ = 6 ÷ 2 = 3 cm.
3. Each wave is a half circle whose diameter is 3 cm, so the compass radius is 3 ÷ 2 = 1.5 cm.
4. Mark M, the midpoint of PY (1.5 cm from P), and N, the midpoint of YQ (4.5 cm from P).
5. Open the compass to 1.5 cm. Put the tip on M and draw the half circle from P to Y above the line.
6. Keep the same opening. Put the tip on N and draw the half circle from Y to Q below the line. The two waves are identical and bulge opposite ways.

In shortAny length of central line works. Split it in half; each half is the diameter of one wave, so the compass radius is one quarter of the whole line. For a 6 cm line: Y at 3 cm, radius 1.5 cm, tips at 1.5 cm and 4.5 cm from P.

Watch this explained “The Wavy Wave — the line decides the radius”, 4:55 into Every curve in these figures is part of a circle · हिंदी में देखें

Open NCERT p. 191One way to think about it

1. There are many correct drawings; here is one. Draw AB = 8 cm and mark X at its middle, so AX = XB = 4 cm. The two waves must stand on equal lengths, or they cannot be identical.
2. A wave flatter than a half circle is a piece of a bigger circle, so its centre is NOT on the line. For the wave that bulges up over AX, the centre goes below the line; for the wave that bulges down over XB, the centre goes above it.
3. The centre must be the same distance from A and from X, so it lies straight below M, the midpoint of AX (2 cm from A). Mark P, 1.5 cm straight below M.
4. Put the compass tip on P and open it to reach A (this opening is 2.5 cm). Draw the arc from A to X above the line. It rises only 2.5 − 1.5 = 1 cm, less than the 2 cm of a half circle.
5. Mark N, the midpoint of XB (6 cm from A), and Q, 1.5 cm straight above N. With the SAME opening of 2.5 cm, tip on Q, draw the arc from X to B below the line.
6. Both arcs use the same opening, and both centres are 1.5 cm from the line, so the two waves are identical and bulge opposite ways.

In shortKeep X in the middle so both waves stand on equal lengths. Put the compass tip off the line, straight below the middle of AX for the upward wave and straight above the middle of XB for the downward wave, both the same distance from the line, and use the same opening for both. Example: AB = 8 cm, tips 1.5 cm from the line, opening 2.5 cm, each wave 1 cm high.

Watch this explained “Arcs shallower than a half circle”, 5:48 into Every curve in these figures is part of a circle · हिंदी में देखें

Open NCERT p. 194One way to think about it

1. There are many correct ways to do this; here is one. Draw the rectangle first with its corners on dots: 8 spaces across and 4 spaces down.
2. Make all four small squares the same size, say 2 spaces by 2 spaces.
3. Place one square outside each corner of the rectangle. The top-left square ends 2 spaces to the left of the rectangle and 1 space above it.
4. Repeat the same gaps at the other corners: the top-right square is 2 spaces to the right and 1 space above, the bottom-left is 2 spaces to the left and 1 space below, and the bottom-right is 2 spaces to the right and 1 space below.
5. Check by folding along the rectangle's upright middle line: the top-left square falls exactly on the top-right one, and the bottom-left on the bottom-right. Fold along the flat middle line: the top squares fall on the bottom ones.

In shortKeep all four squares the same size, and count the gaps: the same sideways gap on the left and the right, and the same up-and-down gap above and below. Example: an 8 by 4 rectangle, 2 by 2 squares, each 2 spaces out sideways and 1 space out up or down.

Open NCERT p. 194Matches NCERT’s answer

1. A square needs S1 (all sides equal) and S2 (all angles 90°). The corners sit on dots, so we can count steps instead of measuring.
2. A: every side goes 4 across and 4 up or down, so all four sides are equal. Each side runs along the slant of the dot squares, at half a right angle (45°) to the rows, so two sides meeting at a corner make 45° + 45° = 90°. A passes both tests, so A is a square.
3. B: every side goes 3 across and 4 up or down, so its sides are equal. But B is only 6 steps wide and 8 steps tall, so it leans: the top and bottom angles are about 74° and the side angles about 106°. B is not a square.
4. C: every side goes 5 across and 4 up or down, so its sides are equal. But C is 10 steps wide and only 8 tall: the top and bottom angles are about 103° and the side angles about 77°. C is not a square.
5. D: the short sides go 2 across and 1 up or down, and the long sides go 2 across and 4 up or down, which is twice as long. Its angles are all 90°, but its sides are not all equal, so D is a rectangle, not a square.

AnswerOnly A is a square. B and C have four equal sides but no right angles, and D has four right angles but unequal sides.

Watch this explained “Sorting four figures by test, not by eye”, 7:52 into The two properties that define a rectangle, and the one more a square needs · हिंदी में देखें

Open NCERT p. 194One way to think about it

1. Any tilted figures with corners on dots will do; here are three. Start each one at a dot and count steps (right, left, up, down) along each side.
2. Square 1: side 1 goes 3 right and 1 up; side 2 goes 1 left and 3 up; side 3 goes 3 left and 1 down; side 4 goes 1 right and 3 down, back to the start.
3. Square 2: 2 right and 2 up; then 2 left and 2 up; then 2 left and 2 down; then 2 right and 2 down, back to the start.
4. Rectangle: 2 right and 1 up; then 2 left and 4 up; then 2 left and 1 down; then 2 right and 4 down, back to the start.
5. Check the squares. Every side is the same slanted step (3 and 1, or 2 and 2), only turned, so all four sides are equal (S1). At each corner the two counts swap and one direction reverses: that is a quarter turn, so each angle is 90° (S2). You can confirm with the corner of a set square.
6. Check the rectangle. Opposite sides are the same steps done backwards, so they are equal (R1). The long side, 2 left and 4 up, is the short side's step (2 right, 1 up) turned a quarter turn and done twice, so every angle is 90° (R2). The long sides are twice the short ones, so it is a rectangle but not a square.

In shortBuild each side from counted steps. For a tilted square, every side uses the same two counts, swapped and turned at each corner (for example 3 right 1 up, then 1 left 3 up). For a tilted rectangle, the second side is the first side's turned step taken a whole number of times (for example 2 right 1 up, then 2 left 4 up). Counting then shows equal sides and right angles.

Watch this explained “Counted, not measured: a square on dots”, 6:21 into Naming corners in order, and why a rotated square is still a square · हिंदी में देखें