Construction — Questions and Answers
Welcome to HSLC Guru. This page gives complete, step-by-step solutions to every Work it Out exercise, the Let us Construct tasks and the worked Examples of ASSEB Class 6 New Mathematics Chapter 8, Construction (অংকন), along with some extra practice questions.
Summary
This chapter teaches how to draw geometrical shapes using a ruler and a compass. Through artwork we learn to draw a circle: if we mark every point that is a fixed distance (say 5 cm) from a point P, we get a circle. The distance from the centre to any point on the circle is the radius, and a line segment through the centre joining two points on the circle is the diameter (diameter = 2 × radius).
Next it covers the properties, naming and rotation of squares and rectangles — a rectangle has equal opposite sides and all angles 90°, while a square has all sides equal and all angles 90°. Using a ruler, protractor and compass we construct a square or rectangle of a given side, divide a rectangle into identical squares, and draw a square inside or outside a circle.
Finally it deals with the diagonals of rectangles and squares, constructing a rectangle from a side and a diagonal, and finding a point equidistant from two given points using a compass (through the rocket construction).
Summary: This page gives complete, step-by-step answers to every Work it Out exercise (8.1, 8.2, 8.3), the Let us Construct tasks and the worked Examples of ASSEB Class 6 New Mathematics Chapter 8, Construction. It covers constructing circles and their parts (centre, radius, diameter), artwork like Rangoli and waves, properties, naming and rotation of squares and rectangles, constructing a square and a rectangle with ruler, protractor and compass, dividing a rectangle into identical squares, diagonals of rectangles and squares, and finding a point equidistant from two given points, with extra MCQs, fill in the blanks, true or false and short answer questions.
Textbook Questions and Answers
8.1 Artwork — the circle and its parts
On a sheet mark a point P. If you mark as many points as possible that are 5 cm away from P in different directions, what figure will you get?
Answer: Marking all points that are 5 cm from P gives a circle of radius 5 cm with centre P. Place the compass pointer on the 0 mark and the pencil on the 5 cm mark of the ruler, then keep the pointer fixed at P and turn the pencil around — this traces the whole circle. Every point on the circle is the same distance (5 cm) from P.
What is a curve? What do the centre, radius and diameter of a circle mean?
Answer: Any shape that can be drawn on paper is a curve (a circle, a wavy line, etc. are all curves). For a circle — the centre is the middle point (P); the radius is the distance from the centre to any point on the circle; and the diameter is a line segment through the centre joining two points on the circle. The diameter is always twice the radius.
Artwork — Rangoli and Waves
How can the Rangoli design be drawn with a ruler and a compass?
Answer: Step 1: Draw a circle of radius 3 cm with centre O and draw two perpendicular diameters. Step 2: Choose the middle point of each radius; taking those middle points as centres, draw four small circles whose diameter equals the radius of the original circle. This gives the flower-like Rangoli. (Drawing one more circle with centre O and radius OA makes the design even prettier.)
To draw waves, a base line AB of length 12 cm is taken and X is marked with AX = 2 cm. If the first wave is a half circle of radius 1 cm, where should Y be for the second wave, and what are the lengths of XY, YB, AY and AB?
Answer: The first wave has diameter 2 cm (ending at X). For the second wave to be identical, its diameter must also be 2 cm, so XY = 2 cm. Hence Y is 2 cm from X (AY = AX + XY = 2 + 2 = 4 cm). The remaining YB = AB − AY = 12 − 4 = 8 cm, and AB = 12 cm.
Work it Out 8.1
1. Take a base line of length 10 cm. Draw two half circles representing identical wavy waves. What will be the radius of each half circle?
Answer: Dividing the 10 cm base line into two equal parts gives 10 ÷ 2 = 5 cm for each part. This 5 cm is the diameter of each half circle, so the radius of each half circle = 5 ÷ 2 = 2.5 cm (that is, 2 cm 5 mm).
2. Try to recreate the figure where the waves are smaller than a half circle. Where should the tip of the compass be placed to get two identical waves of this type?
Answer: To make a wave smaller than a half circle (an arc of less than 180°), the compass tip must be placed not on the base line but on the perpendicular bisector of each equal part, below the base line, with the radius greater than half of that part. For the two waves to be identical, the tip must be placed at the same distance and with the same radius for both parts.
3. Try to draw the above wavy waves with base lines of different lengths.
Answer: For a base line of any length L, dividing it into two equal parts makes the radius of each semicircular wave = L ÷ 4. For example, an 8 cm base line gives radius 2 cm, a 12 cm base line gives radius 3 cm, and so on.
4. Draw four identical waves with half circles.
Answer: Divide the base line into four equal parts. Each part becomes the diameter of a half circle, so the radius = base length ÷ 8. For example, on a 12 cm base line each part is 3 cm and each wave has radius 1.5 cm.
8.2 Squares and Rectangles
What are the properties of a square and a rectangle, and how are these shapes named?
Answer: For a rectangle — opposite sides are equal in length and all angles are 90°. For a square — all sides are equal in length and all angles are 90°. Naming: start at one vertex (say A) and walk along the border; name the vertices in the order you reach them — ABCD, or in the opposite direction ADCB. You cannot name it ACBD or ABDC.
Work it Out 8.2
1. What is the name of the adjacent square? (A top-left, B top-right, C bottom-right, D bottom-left) — (a) ABDC (b) ABCD (c) ACDB (d) ADBC
Answer: (b) ABCD. Walking along the border from A gives A → B → C → D, so the square is named ABCD.
Work it Out 8.3
1. Can you say whether a rotated rectangle is also a rectangle?
Answer: Yes. Rotating a shape does not change its side lengths or its angles. So after rotation the opposite sides are still equal and all angles are still 90° — hence a rotated rectangle is still a rectangle.
2. Draw a rectangle ABCD. Fixing the corner at A, draw three rotated rectangles and check whether they satisfy the properties of a rectangle.
Answer: Keeping A fixed and drawing the rectangle turned through different angles, each figure still has equal opposite sides and all angles 90°. So every rotated figure satisfies the rectangle properties — they are all rectangles.
3. Your geometry box has two set squares — a 30°-60°-90° set square and a 45°-45°-90° set square. (a) With a pair of 30°-60°-90° set squares placed as shown, in the quadrilateral formed: (i) the measure of each angle, (ii) the measure of the opposite sides, (iii) the name of the quadrilateral, (iv) any other properties. (b) With a pair of 45°-45°-90° set squares: (i) are all sides equal, (ii) what about the angles, (iii) which name do you suggest, (iv) any properties?
Answer: Joining two congruent right-angled triangles along their hypotenuse gives a quadrilateral.
- (a) 30°-60°-90° set squares: (i) each angle is 90°. (ii) opposite sides are equal. (iii) it is a rectangle. (iv) the diagonals are equal and opposite sides are parallel.
- (b) 45°-45°-90° set squares: (i) yes, all sides are equal. (ii) all angles are 90°. (iii) it is a square. (iv) the diagonals are equal and perpendicular to each other; a square is a special kind of rectangle.
4. Ratul’s home is at point A. He moves 6 km East, then 4 km South, then 3 km West and finally 4 km North. Trace his path. (i) Where is he finally located? (ii) What is the shape of the quadrilateral he moved around? (iii) If 3 km is replaced by 4 km, what shape will it be? (Hint: take 1 km = 1 cm.)
Answer: Taking 1 km = 1 cm and starting at A(0, 0) — East 6 → (6, 0); South 4 → (6, −4); West 3 → (3, −4); North 4 → (3, 0).
- (i) The 4 km South and 4 km North cancel, and East 6 − West 3 = 3 km remains. So he does not return home; he ends up 3 km East of A.
- (ii) Joining his end point back towards A, the four-sided outline has two parallel horizontal sides of 6 km and 3 km — which are unequal. So it is not a rectangle; it is a trapezium (a quadrilateral with one pair of parallel but unequal sides). It would be a rectangle only if the East and West distances were equal.
- (iii) Replacing 3 km with 4 km makes the parallel sides 6 km and 4 km — still unequal, so it is still a trapezium (and he would end 2 km East of A).
8.3 Construction of Squares and Rectangles — square ABCD (side 5 cm)
How can a square ABCD of side 5 cm be constructed?
Answer: Follow these steps —
- Step 1: Using a ruler, draw a line segment AB of length 5 cm.
- Step 2: At A, place the protractor with its base line along AB, mark 90° and join it to A — this gives a perpendicular to AB at A.
- Step 3: On the perpendicular mark D so that AD = 5 cm (using a ruler, or set the compass to 5 cm, place its point at A and cut the perpendicular at D).
- Step 4: Draw another perpendicular to AB at B, as in Step 2.
- Step 5: With the compass at the same 5 cm setting, place its point at B and mark C on the perpendicular so that BC = 5 cm.
- Step 6: Join C and D with a ruler to get the square ABCD of side 5 cm. Verify: DC = 5 cm, ∠D = 90°, ∠C = 90°.
Let us Construct (squares and rectangles)
1. Draw a rectangle of sides 3 cm and 5 cm. Does it satisfy the properties of a rectangle? Verify.
Answer: Using the same steps, take AB = 5 cm, draw perpendiculars at A and B, mark AD = BC = 3 cm on them and join CD to get a 5 cm × 3 cm rectangle. On measuring, opposite sides are equal (5 cm and 3 cm) and all four angles are 90° — so the rectangle properties are satisfied.
2. Draw a square of side 4 cm and verify whether the properties of a square are satisfied.
Answer: Repeat the six steps with 4 cm instead of 5 cm to get a square of side 4 cm. On measuring, all sides are 4 cm and all angles are 90° — so the square properties are satisfied.
3. Can you draw a quadrilateral whose angles are all 90° but whose opposite sides are not equal?
Answer: No. If all four angles of a quadrilateral are 90° it must be a rectangle, and a rectangle always has equal opposite sides. So a quadrilateral with all angles 90° but unequal opposite sides cannot be drawn.
4. Can you draw a quadrilateral whose sides are all equal but whose angles are not necessarily 90°?
Answer: Yes. A quadrilateral with all sides equal but angles not 90° is a rhombus. It looks like a slanted (tilted) square — all four sides are equal but the angles are not right angles.
Constructing a rectangle divided into identical squares
How can a rectangle be constructed that can be divided into two identical squares?
Answer: Step 1: Draw a line segment AF without measuring its length. Step 2: Construct a perpendicular to AF at A. Step 3: With a compass mark B on the perpendicular so that AB = AF. Step 4: With the same compass setting, put the point at B and mark C so that AB = BC. Step 5: Do the same from F to mark E and D. Step 6: Join BE and CD to get the rectangle ACDF, which breaks into two identical squares ABEF and BCDE.
Let us Construct (identical squares)
1. With the same idea, construct a rectangle that can be divided into three identical squares.
Answer: Take one side (say AF = 3 cm) and, on the perpendicular, mark that same length three times in a row (total 9 cm). This gives a rectangle of length 9 cm and breadth 3 cm, which divides into three identical 3 cm × 3 cm squares.
2. Can all rectangles be divided into some number of identical squares? If not, give the side lengths of a rectangle that cannot be divided into (a) two identical squares, (b) three identical squares.
Answer: No — a rectangle can be divided into identical squares only when its length is an exact multiple of its breadth. (a) For two identical squares the length must be exactly twice the breadth; so a rectangle of length 5 cm and breadth 2 cm (5 ≠ 2 × 2) cannot be divided into two identical squares. (b) For three identical squares the length must be exactly three times the breadth; so a rectangle of length 7 cm and breadth 2 cm (7 ≠ 3 × 2) cannot be divided into three identical squares.
Square within a circle
How can a square be constructed inside a circle?
Answer: Draw two diameters of the circle that are perpendicular to each other. Join the four end points of these diameters (which lie on the circle) in order — this gives a square inside the circle.
8.5 Diagonals of Rectangles and Squares
What is a diagonal of a rectangle or square? What do you find on measuring the two diagonals?
Answer: In rectangle ABCD, the line segments joining the opposite vertices A & C and B & D are the diagonals AC and BD. On measuring, the two diagonals of a rectangle (or square) are equal in length. A diagonal also divides each pair of opposite angles into two smaller angles.
Example 1
Construct a rectangle in which one of the diagonals divides opposite angles into 40° and 50°.
Answer: Step 1: Draw a line AB of any length (A and B are the first and second vertices). Step 2: With a protractor, construct an angle of 40° at A and draw ray AE. Step 3: At B draw a perpendicular BC to AB; the point where BC meets AE is the third vertex C. Step 4: At A draw another perpendicular AF to AB. Step 5: Draw a perpendicular to BC at C; it meets AF at D, the fourth vertex. So rectangle ABCD is obtained. Measuring ∠CAD gives 50° (because ∠A = 90° and ∠BAC = 40°, so ∠CAD = 90° − 40° = 50°).
Example 2
Construct a rectangle in which one side is 4 cm and one diagonal is 5 cm.
Answer: Step 1: Draw a line segment AB of length 4 cm. Step 2: At B draw a line perpendicular to AB; the third vertex C lies somewhere on it. Step 3: Since the diagonal is 5 cm, set the compass to 5 cm, place its point at A and draw an arc that cuts the perpendicular at C (AC = 5 cm diagonal); then BC = $\sqrt{5^2 – 4^2} = \sqrt{25 – 16} = \sqrt{9} = 3$ cm. Step 4: Draw a perpendicular to AB at A and a perpendicular to BC at C; they meet at D, the fourth vertex. So rectangle ABCD of sides 4 cm × 3 cm with a 5 cm diagonal is obtained.
Let us Construct (diagonals)
1. Construct a rectangle in which one of the diagonals divides the opposite angles into (a) 20° and 70° (b) 30° and 60°.
Answer: Use the method of Example 1 — (a) make a 20° angle at A, draw the ray, draw the perpendicular at B to get C, and then the fourth vertex D; the remaining angle ∠CAD = 90° − 20° = 70°. (b) similarly, a 30° angle at A gives the remaining angle ∠CAD = 90° − 30° = 60°. (In each case the construction works because the two angles add up to 90°.)
8.6 Equidistant Point from Two Given Points (rocket construction)
Using a compass, how can we find a point A that is the same distance (4 cm) from two given points B and C? (through the rocket construction)
Answer: The middle part of the rocket, DECB, is a rectangle (BC = DE = 4 cm, BD = CE = 6 cm). To find the point A that is 4 cm from both B and C —
- Method 1: Draw a circle of radius 4 cm with centre B and another circle of radius 4 cm with centre C. The two circles cut each other at two points; the point above the rectangle is the required point A.
- Method 2: Instead of full circles, draw two arcs of radius 4 cm from B and from C; where the arcs cross is A. Join AB and AC and check — both are 4 cm.
In the same way, to find the point P that is 3 cm from both D and E, draw two arcs (or circles) of radius 3 cm with centres D and E; their intersection inside the rectangle is P. Finally join A-B, A-C, P-D and P-E to complete the rocket.
Let us Construct (equidistant point)
1. Construct a square with all sides 6 cm. Identify its diagonals. Can you measure the diagonals with a ruler? How would you draw a line of the same length as a diagonal?
Answer: The diagonal of a 6 cm square = $6\sqrt{2} \approx 8.49$ cm, so it can be measured with a ruler (about 8 cm 5 mm). But to draw a line of exactly the same length, it is better to use a compass — set the compass to the length of the diagonal (both tips on its two ends), then transfer that length to a line by cutting an arc. This gives a line equal to the diagonal.
2. Is there a 4-sided figure whose opposite sides are equal but which is not a rectangle? If it exists, can you construct it?
Answer: Yes — it is a parallelogram. Following the hint: take two sides of 6 cm and 4 cm; draw an arc of radius 4 cm with centre C and an arc of radius 6 cm with centre A; their intersection is D. Then AD and CD equal BC and AB respectively, so the opposite sides are equal but the angles are not 90° — hence it is not a rectangle.
3. Using the earlier ideas, construct a cylinder (2-D figure).
Answer: Following the hint: take a line PQ of length 4 cm. Choose two suitable points A and B (above and below PQ) as the compass positions, and draw two curves (arcs) that meet each other neatly at P and Q. Take another line RS at a suitable distance from PQ and draw curves in the same way. Finally rub out the lines PQ and RS to get the shape of a cylinder.
Additional Questions and Answers
Multiple Choice Questions (MCQ)
1. The distance from the centre of a circle to any point on the circle is called the — (a) diameter (b) radius (c) diagonal (d) arc
Answer: (b) radius
2. A line segment through the centre joining two points on the circle is called the — (a) radius (b) arc (c) diameter (d) side
Answer: (c) diameter
3. Each angle of a square measures — (a) 60° (b) 90° (c) 100° (d) 45°
Answer: (b) 90°
4. The diagonal of a square of side 6 cm is about — (a) 6 cm (b) 8.49 cm (c) 12 cm (d) 36 cm
Answer: (b) 8.49 cm
5. If one side of a rectangle is 4 cm and its diagonal is 5 cm, the other side is — (a) 1 cm (b) 2 cm (c) 3 cm (d) 9 cm
Answer: (c) 3 cm
6. To find a point equidistant from two given points, it is convenient to use a — (a) ruler only (b) compass (c) pencil (d) eraser
Answer: (b) compass
7. The opposite sides of a rectangle are — (a) unequal (b) equal (c) perpendicular (d) curved
Answer: (b) equal
8. Joining two congruent 45°-45°-90° set squares along the hypotenuse gives a — (a) rectangle (b) square (c) triangle (d) parallelogram
Answer: (b) square
9. Joining two congruent 30°-60°-90° set squares along the hypotenuse gives a — (a) square (b) rectangle (c) circle (d) trapezium
Answer: (b) rectangle
10. Two identical semicircular waves on a 10 cm base line each have radius — (a) 5 cm (b) 2.5 cm (c) 10 cm (d) 1 cm
Answer: (b) 2.5 cm
Fill in the Blanks
- All points on a circle are at ______ distance from the centre. (equal)
- Diameter of a circle = 2 × ______. (radius)
- All ______ of a square are equal. (sides)
- Each angle of a rectangle is ______. (90°)
- A line segment joining the opposite vertices of a rectangle or square is called a ______. (diagonal)
True or False
- The diameter of a circle is twice its radius. — True
- A rotated rectangle is no longer a rectangle. — False (it remains a rectangle after rotation)
- All sides and all angles of a square are equal. — True
- If all angles of a quadrilateral are 90°, its opposite sides are always equal. — True
- If two arcs of equal radius are drawn from two points, their intersection is equidistant from both points. — True
Short Answer Questions
1. Define the radius and the diameter of a circle.
Answer: The radius is the distance from the centre to any point on the circle. The diameter is a line segment through the centre joining two points on the circle (diameter = 2 × radius).
2. What is the difference between a square and a rectangle?
Answer: Both have all angles 90°. But a square has all four sides equal, while a rectangle has only its opposite sides equal (the adjacent sides may differ). So every square is a rectangle, but every rectangle is not a square.
3. Find the length of the diagonal of a square of side 6 cm.
Answer: Diagonal = $\sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \approx 8.49$ cm.
4. How do you find a point equidistant from two given points using a compass?
Answer: Draw arcs (or circles) of the same radius taking each given point as the centre. The point where the two arcs intersect is at an equal distance from both given points — that is the required equidistant point.
Key Terms
| Term | Meaning |
|---|---|
| Construction | Drawing geometrical shapes with a ruler and a compass |
| Compass | Instrument used to draw circles and arcs |
| Protractor | Instrument used to measure and draw angles |
| Circle | A curve made of points that are the same distance from the centre |
| Radius | Distance from the centre to a point on the circle |
| Diameter | A chord through the centre (= 2 × radius) |
| Diagonal | A line segment joining opposite vertices of a figure |
| Perpendicular | A line standing at 90° to another line |
| Semicircle | Half of a circle |
| Equidistant point | A point that is the same distance from two given points |
| Vertex | A corner point of a figure |