6ನೇ ತರಗತಿ ಗಣಿತ (ಗಣಿತ ಪ್ರಕಾಶ) ಭಾಗ-2ರ 8ನೇ ಅಧ್ಯಾಯ “ರಚನೆಗಳೊಂದಿಗೆ ಆಟ” (Playing with Constructions) ಪಾಠದ ಸರಳ ಇಂಗ್ಲಿಷ್ ಟಿಪ್ಪಣಿಗಳು ಇಲ್ಲಿವೆ. ಕೈವಾರ ಮತ್ತು ಅಳತೆಪಟ್ಟಿ ಬಳಸಿ ವೃತ್ತ, ಚೌಕ ಮತ್ತು ಆಯತಗಳ ರಚನೆ, ಕರ್ಣಗಳು, ಒರಟು ಚಿತ್ರದ ಮೂಲಕ ಯೋಜನೆ ಹಾಗೂ ಎರಡು ಬಿಂದುಗಳಿಂದ ಸಮಾನ ದೂರದ ಬಿಂದುವನ್ನು ಕಂಡುಹಿಡಿಯುವುದನ್ನು ಪಠ್ಯಪುಸ್ತಕದ ಚಿತ್ರಗಳೊಂದಿಗೆ ವಿವರಿಸಲಾಗಿದೆ. ಎಲ್ಲಾ “Figure it Out” ಮತ್ತು “Construct” ಪ್ರಶ್ನೆಗಳಿಗೆ ಉತ್ತರಗಳು, ಆಟಗಳು, ಉಚಿತ PDF ಗಳು ಮತ್ತು ಶಾಲೆಯ ಹೆಸರಿನಲ್ಲಿ ಬಳಸಬಹುದಾದ ಪ್ರಶ್ನೆಪತ್ರಿಕೆ (Word) ಲಭ್ಯವಿದೆ.
Playing with Constructions
The full lesson in simple words with textbook maps and pictures, animations, games and quizzes, followed by answers to every question and activity.
Questions and answers: all Figure it Out questions, puzzles and activities.
House construction
Falling squaresThe Lesson
With a ruler and a compass we can draw beautiful figures, squares and rectangles. Let us construct!
8.1 Artwork
Observe the figures in Fig. 8.1 and try drawing them freehand. Then use a ruler and a compass.
Curves are any shapes that can be drawn on paper with a pencil; they include straight lines, circles and other figures.
Mark a point P. Mark many points that are 4 cm away from P. Open the compass against a ruler so that the tip and the pencil are 4 cm apart. Keep the tip fixed and move only the pencil: the curve is a circle.
Every point on the circle is 4 cm from P. P is the centre of the circle, and the distance between the centre and any point on the circle is the radius.
Fig. 8.1 Artwork
Fig. 8.2 Opening the compass to 4 cm; centre and radiusConstruct: A Person, Wavy Wave, Eyes
A Person has two parts. For the curved top of the body, fix a radius in the compass and try placing its tip in different places to find which point gives the curve.
Wavy Wave: take the central line AB = 8 cm. The first wave is a half circle.
Eyes: the upper and lower curves of each eye are drawn the same way as the curve in ‘A Person’ (see the hint at the end).
A Person
Drawing the curved top with a compass
Wavy Wave
The first wave as a half circle on AB = 8 cm
Eyes8.2 Squares and rectangles
In rectangle ABCD, A, B, C and D are the corners; AB, BC, CD and DA are the sides; ∠A, ∠B, ∠C and ∠D are the angles. AB and CD are opposite sides, and so are AD and BC.
- R1 In a rectangle, the opposite sides are equal in length.
- R2 All the angles are 90°.
- S1 In a square, all the sides are equal.
- S2 All the angles are 90°.
Fig. 8.3 Squares and a rectangleRectangle ABCD can also be named BCDA, CDAB, DABC, ADCB, DCBA, CBAD or BADC, but not ABDC or ACBD. In a valid name, the corners occur in an order of travel around the figure, starting from any corner.
Rotated squares and rectangles: rotating a square does not change its lengths and angles, so it is still a square. A rotated rectangle is still a rectangle.
A square rotated is still a square8.3 Constructing squares and rectangles
To construct square PQRS of side 6 cm:
- Draw PQ = 6 cm.
- Using a protractor, mark a point to draw a perpendicular to PQ through P.
- Mark S on the perpendicular so that PS = 6 cm (with a ruler, or with a compass opened to 6 cm).
- Draw a perpendicular to PQ through Q.
- Mark R on it so that QR = 6 cm (easy with the compass).
- Join RS. RS = 6 cm and ∠R = ∠S = 90°.
Steps 1–3: square PQRS of side 6 cm
Steps 4–68.4 An exploration in rectangles
Construct rectangle ABCD with AB = 7 cm and BC = 4 cm. Point X moves along AD and point Y moves along BC. Measure XY for different positions and record them in a table.
Points X on AD and Y on BCBreaking rectangles
To construct a rectangle that can be divided into two identical squares, first draw a rough diagram. Since ABEF and BCDE are identical squares, AF = AB = BE = FE and BE = BC = CD = ED. Equal sides are shown by putting a ‘|’ mark on them.
If AF = 4 cm, then AC = 8 cm. The length AF can also be transferred with a compass, without using a ruler, to mark B and C.
Rough figure: two identical squares
Transferring the length AF with a compassMore constructions
Square within a rectangle, Falling Squares, Shadings, Square with a Hole, Square with more Holes and Square with Curves.
A square within a rectangle
Falling squares (4 cm)Squares of side 3, 5 and 7 cm
Shadings
Square with a hole
Square with more holes
Square with curves8.5 Exploring diagonals of rectangles and squares
In rectangle PQRS, PR and QS are the diagonals. The right angles at P and R are opposite angles; so are those at Q and S.
A diagonal divides each of a pair of opposite angles into two smaller angles: PR divides ∠R into g and h, and ∠P into c and d.
Rectangle PQRS with diagonalsConstructing a rectangle when an angle is given
Example 1: a diagonal divides the opposite angles into 60° and 30°.
- Draw AB of any length and a perpendicular at B.
- Draw a line from A at 60° to AB; it meets the perpendicular at C.
- Draw a line through A perpendicular to AB. ∠A is now divided into 60° and 30°.
- Find D: draw a perpendicular to BC at C (Method 1), or mark D with a compass so that AD = BC (Method 2). Join CD.
Rough diagram: 60° and 30°
Step 4, Method 1
Step 4, Method 2: AD = BC with a compassConstructing a rectangle when a side and a diagonal are given
Example 2: one side 5 cm and a diagonal 7 cm.
- Draw DC = 5 cm.
- Draw line l perpendicular to DC at C. B lies on l.
- B is 7 cm from D: draw a circle (or just an arc) of radius 7 cm with centre D. Where it cuts l is B.
- Draw perpendiculars to DC at D and to BC at B; they meet at A. Check that ABCD satisfies R1 and R2.
Rough diagram: side 5 cm, diagonal 7 cm
Circle of radius 7 cm with centre D
Completing the rectangle8.6 Points equidistant from two given points
House: all the lines forming the border are 5 cm.
- Draw DE = 5 cm, the perpendiculars DB = EC = 5 cm, and the door (1 cm × 2 cm).
- Draw a circle of radius 5 cm with centre B.
- Draw a circle of radius 5 cm with centre C. The point where the two circles meet is 5 cm from both B and C: it is A. (Just two arcs are enough.) Join AB and AC.
- With radius 5 cm and centre A, draw the arc from B to C. The house is ready!
House: all border lines 5 cm
Step 1
Step 2: circle with centre B
Step 3: circles with centres B and C
Point A found with arcs
The house is ready!★ Summary
- All the points of a circle are at the same distance from its centre. This distance is called the radius.
- A compass can be used to construct circles and their parts.
- A rough diagram can be useful in planning how to construct a given figure.
- A rectangle can be constructed given the lengths of its sides, or one side and a diagonal.
Game 1: Square, rectangle or neither?
Read the clue and tap the answer. 8 cards!
Game 2: Quick construction quiz
- Radius for a half-circle wave on AB = 8 cm (two waves)
- Which is NOT a name for square SPQR?
- A rectangle split into 2 identical squares has AF = 4 cm. AC = ?
- Square within an 8 cm × 4 cm rectangle: side of the square?
- XY when X and Y are both 1 cm from A and B (AB = 7 cm)
- The diagonals of a rectangle are
Game 3: Match the words
Questions & Answers
Figure it Out, Construct and in-text questions pages 59–88
8.1 Artwork (pages 59–64)
Imagine marking all the points 4 cm from P. How would they look?
Page 60They form a circle with centre P and radius 4 cm.
Take a point on the circle. What will be its distance from P: equal to, less than or greater than 4 cm? What about another point?
Page 61Equal to 4 cm. Every point on the circle is 4 cm from P.
Recreate the images in Fig. 8.1. Has the use of instruments made the construction easier?
Page 61Yes. With a compass the circles are perfectly round and of the exact size we want. Fix the radius, place the tip at the centre and turn the compass.
A Person: how will you draw this?
Page 62Draw the head with a compass and the neck with a ruler. Draw the three straight sides of the body with a ruler. For the curved top, fix a radius and try placing the compass tip at different points above the body until the arc passes through both top corners.
Wavy Wave (AB = 8 cm): What radius should be taken to get the half circle? What should be the length of AX?
Page 63There are two equal half circles on AB, so each has a diameter of 8 ÷ 2 = 4 cm. Radius = 2 cm and AX = 4 cm.
Take a central line of a different length and draw the wave on it.
Page 63Example: AB = 6 cm. Mark X at 3 cm. Draw a half circle of radius 1.5 cm above AX and another below XB.
Recreate the figure where the waves are smaller than a half circle.
Page 63Use the same method as for ‘A Person’: take a radius larger than half of AX and keep the compass tip farther from AX (below the line for the upper wave and above it for the lower wave). Use the same radius and the same distance for both waves so they are identical.
Eyes: how do you draw these eyes with a compass?
Page 64
Hint: drawing the eye with points A and BDraw a light horizontal supporting line. Place the compass tip at a point B below the line to draw the upper curve, and at a point A above the line (the same distance away) to draw the lower curve, so that the two curves together are symmetrical. Draw the pupil as a small shaded circle. Repeat for the second eye.
8.2 Squares and rectangles (pages 64–66)
Which of the following is not a name for square SPQR? 1. PQSR 2. SPQR 3. RSPQ 4. QRSP
Page 65
Square SPQR1. PQSR is not a name: going from Q to S is a diagonal, not a side. The corners must be in order of travel around the square.
Draw the rectangle and four squares configuration (Fig. 8.3) on dot paper. What did you do to place the four squares symmetrically?
Page 66Draw the rectangle in the middle. Place each square at the same number of dots away from its nearest corner of the rectangle, both across and up/down, so all four squares are placed alike.
Identify if there are any squares in the collection A, B, C, D.
Page 66
Figures A, B, C, D on a dot grid- A is a square: all four sides go 4 dots across and 4 dots up/down, so all sides are equal and all angles are 90°.
- B is not a square: its sides are equal but it is 3 dots wide and 4 dots high on each side, so the angles are not 90°.
- C is not a square: 5 dots across and 4 up/down; the angles are not 90°.
- D is a rectangle but not a square: one pair of sides is shorter than the other pair.
Think: can we reason out if the sides are equal and the angles are right only by looking at the corners on the dot grid?
Page 66Yes. Count how many dots each side moves across and up/down. Sides that move the same counts are equal. If one side moves (4 across, 4 up) and the next moves (4 across, 4 down), the counts are swapped and the corner is a right angle.
Draw at least 3 rotated squares and rectangles on a dot grid with corners on the dots and verify their properties.
Page 66Example square: corners move 2 across and 1 up, then 1 across and 2 down, and so on. Example rectangle: sides of (2 across, 1 up) and (2 back, 4 up) as in D. Check S1, S2 or R1, R2 by counting dots and measuring.
8.3 Constructing squares and rectangles (pages 67–69)
Why should PS be 6 cm long? How long is RS and what are ∠R and ∠S?
Page 68PS must be 6 cm because all sides of a square are equal (S1). RS = 6 cm, and ∠R = ∠S = 90°.
Draw a rectangle with sides 4 cm and 6 cm and check both rectangle properties.
Page 69Draw AB = 6 cm, perpendiculars at A and B, mark AD = BC = 4 cm, join DC. Check: AB = DC = 6 cm and AD = BC = 4 cm (R1); all angles measure 90° (R2).
Draw a rectangle of sides 2 cm and 10 cm and check both properties.
Page 69Draw AB = 10 cm, perpendiculars at A and B, mark AD = BC = 2 cm, join DC. Opposite sides are 10 cm and 2 cm (R1) and all angles are 90° (R2).
Is it possible to construct a 4-sided figure in which all the angles are 90° but opposite sides are not equal?
Page 69No. When all four angles are 90°, the opposite sides always come out equal, so the figure is a rectangle.
8.4 An exploration in rectangles (pages 69–75)
At which positions will X and Y be closest? Farthest? How does the minimum distance compare with AB?
Page 70Closest: when X and Y are the same distance from A and B (XY is parallel to AB); then XY = 7 cm, the same as AB. Farthest: when X = A and Y = C, or X = D and Y = B; then XY is a diagonal.
Find XY when X is 5 mm from A and Y is 3 cm from B; X 1 cm and Y 1 cm; X 2 cm and Y 4 cm. Record in a table.
Page 70By construction and measurement: XY ≈ 7 cm 4 mm; 7 cm 0 mm; ≈ 7 cm 3 mm.
When X and Y are the same distance from A and B (5 mm, 1 cm, 1 cm 5 mm …), how does XY compare with AB? What is the shape of ABYX?
Page 71XY = AB = 7 cm every time, and ABYX is a rectangle.
How does the farthest distance between X and Y compare with AC and BD?
Page 71The farthest distance is equal to AC and BD (about 8 cm 1 mm): the diagonals.
Breaking rectangles: if AF = 4 cm (two identical squares), what must AC be? Complete the rectangle.
Page 72AC = 8 cm (AB = BC = AF = 4 cm). Draw AF, a perpendicular at A, transfer AF with the compass to mark B and then C, and complete the rectangle with perpendiculars.
Construct a rectangle that can be divided into three identical squares.
Page 73
Rectangle divided into 3 identical squaresDraw AF of any length, say 4 cm. Draw a perpendicular at A and use the compass (radius AF) three times to mark B, C and the last corner, so the long side is 3 × 4 = 12 cm. Complete the rectangle and join the dividing lines.
Give the lengths of the sides of a rectangle that cannot be divided into two identical squares; three identical squares.
Page 73Two squares need length = 2 × breadth, so e.g. 4 cm × 6 cm cannot. Three squares need length = 3 × breadth, so e.g. 4 cm × 10 cm cannot.
A square within a rectangle (8 cm × 4 cm) with the same centre: what is the side of the square and its distance from the outer rectangle?
Page 73The square has side 4 cm (the height of the rectangle). Its sides are (8 − 4) ÷ 2 = 2 cm from each short side of the rectangle. Mark 2 cm from each end of both long sides and join.
Falling squares: construct the 4 cm squares and then the squares of side 3 cm, 5 cm and 7 cm.
Page 74Draw the first square. Each next square starts at the top-right corner of the previous one, its bottom side on the top side of the previous square, aligned as shown. Use the perpendicular method for each square.
Shadings, Square with a Hole, Square with more Holes, Square with Curves.
Page 75- Shadings: construct a big square, divide it into equal smaller squares, draw the diagonals shown and shade.
- Hole: the centre of the circle is where the diagonals of the square meet; use a radius less than half the side.
- More holes: divide the square into 4 equal squares; draw a circle at the centre of each.
- Curves (8 cm square): place the compass tip outside the square on the line through the middle of each side, at the same distance for all four sides, so that each arc passes through two corners and bulges equally.
8.5 Diagonals (pages 75–83)
Compare the lengths of the diagonals PR and QS.
Page 75They are equal.
Are g and h equal? Are c and d equal? Identify pairs of equal angles.
Page 76In a rectangle that is not a square, g ≠ h and c ≠ d. Equal pairs: d = e = h = a and c = b = g = f.
How should the rectangle be constructed so that the diagonal divides the opposite angles into equal parts? What laws did you observe?
Page 76All four sides must be equal: it must be a square. Then each diagonal divides the angles into 45° and 45°. Laws: the diagonals of a rectangle are equal; a diagonal divides opposite angles into equal parts only in a square.
In Example 1 (60° and 30°), what is the next point after AB? Check the other angle at A.
Page 77The next point is C, where the 60° line from A meets the perpendicular at B. The other angle at A is 30° (90° − 60°).
Construct a rectangle in which a diagonal divides the opposite angles into 50° and 40°.
Page 83Draw AB of any length and a perpendicular at B. At A draw a line at 50° to AB; it meets the perpendicular at C. Draw a perpendicular to AB at A, then a perpendicular to BC at C to get D (or mark AD = BC). ∠CAD measures 40°.
Construct a rectangle in which a diagonal divides the opposite angles into 45° and 45°. What do you observe about the sides?
Page 83Construct as above with a 45° line. All four sides come out equal: the rectangle is a square.
Construct a rectangle one of whose sides is 4 cm and the diagonal is 8 cm.
Page 83Draw DC = 4 cm and a perpendicular l at C. With centre D and radius 8 cm, cut l at B. Draw perpendiculars at D and B to meet at A. The other side measures about 6.9 cm.
Construct a rectangle one of whose sides is 3 cm and the diagonal is 7 cm.
Page 83Draw DC = 3 cm and a perpendicular l at C. With centre D and radius 7 cm, cut l at B. Complete with perpendiculars at D and B. The other side measures about 6.3 cm.
8.6 Points equidistant from two points (pages 83–88)
House: where do the two circles meet? How far is it from B and from C? Was it necessary to draw two full circles?
Page 85They meet at A, which is 5 cm from B and 5 cm from C. No, only small arcs from B and C near A are needed.
Construct a bigger house in which all the sides are of length 7 cm.
Page 87Follow the same steps with 7 cm: draw DE = 7 cm, DB = EC = 7 cm, arcs of radius 7 cm from B and C meet at A, join AB and AC, and with centre A and radius 7 cm draw the arc from B to C.
Recreate ‘A Person’, ‘Wavy Wave’ and ‘Eyes’ using ideas from the ‘House’ construction.
Page 87For each curve that joins two points, find the compass point that is the same distance from both end points, using two arcs of equal radius, as point A was found in the house. Then draw the curve with that point as centre.
Is there a 4-sided figure in which all the sides are equal but which is not a square? If so, construct it.
Page 87
Hint: AB = AC = 5 cmYes. Draw AC = 5 cm and AB = 5 cm at an angle that is not 90° (figures B and C on the dot grid are like this). With radius 5 cm draw arcs from B and C; they meet at D. Join BD and CD. All sides are 5 cm but the angles are not 90°.
