ICSE Class 9 > Class 9 Notes > Mathematics > Chapter Notes: Construction of Polygons
Chapter Notes: Construction of Polygons
Introduction
Geometry is like a treasure hunt where precision and creativity meet! In this exciting chapter, we dive into the art of constructing polygons using just a ruler and compass. From quadrilaterals to regular hexagons, you'll learn how to bring shapes to life with simple tools and clear steps. Whether it's drawing a parallelogram with specific sides or crafting a perfect square, this chapter unlocks the secrets of geometric construction, making it both fun and rewarding. Get ready to sharpen your skills and create polygons with confidence!
Construction of Quadrilaterals
- A quadrilateral is constructed by locating its four vertices.
- Always start with a rough free-hand sketch to plan the construction.
- Different methods are used based on given measurements like sides, angles, or diagonals.
Case 1: Four Sides and One Angle
- Draw one side as the base.
- Construct the given angle at one end of the base.
- Measure the adjacent side from the angle.
- Use the remaining sides as radii to draw arcs from two vertices, intersecting at the fourth vertex.
- Join the vertices to complete the quadrilateral.
Example: Construct quadrilateral ABCD with AB = 3.5 cm, BC = 4.0 cm, CD = 5.0 cm, DA = 4.0 cm, and ∠B = 45°.
- Draw BC = 4.0 cm.
- At B, draw BP such that ∠PBC = 45°.
- From BP, measure BA = 3.5 cm.
- With A as center and radius 4.0 cm, and C as center with radius 5.0 cm, draw arcs to intersect at D.
- Join AD and CD to form quadrilateral ABCD.
Case 2: Three Sides and Two Consecutive Angles
- Draw one side as the base.
- Construct the first given angle at one end of the base and measure the adjacent side.
- Construct the second angle at the other end of the base.
- Draw an arc from one vertex with the third side's length to intersect the angle line at the fourth vertex.
- Join the vertices to complete the quadrilateral.
Example: Construct quadrilateral ABCD with AB = 4.0 cm, BC = 4.5 cm, CD = 4.7 cm, ∠B = 60°, and ∠A = 120°.
- Draw BC = 4.5 cm.
- At B, construct ∠MBC = 60° and measure BA = 4.0 cm along BM.
- At A, draw AP such that ∠BAP = 120°.
- With C as center and radius 4.7 cm, draw an arc to intersect AP at D.
- Join C to D to form quadrilateral ABCD.
Case 3: Four Sides and One Diagonal
- Construct a triangle using three sides, including the given diagonal.
- Construct another triangle using the remaining two sides and the diagonal.
- Join the appropriate vertices to complete the quadrilateral.
Example: Construct quadrilateral ABCD with AB = 3.0 cm, BC = 4.0 cm, CD = 5.0 cm, DA = 3.5 cm, and diagonal AC = 4.5 cm.
- Construct triangle ABC using AB = 3.0 cm, BC = 4.0 cm, and AC = 4.5 cm.
- Construct triangle ADC using AD = 3.5 cm, DC = 5.0 cm, and AC = 4.5 cm.
- Join the vertices to form quadrilateral ABCD.
Case 4: Three Sides and Two Diagonals
- Construct a triangle using two sides and one diagonal.
- Construct another triangle using the third side and the other diagonal.
- Join the vertices to complete the quadrilateral.
Example: Construct quadrilateral ABCD with AB = 3.5 cm, BC = 3.0 cm, AD = 3.8 cm, diagonal AC = 5.0 cm, and diagonal BD = 4.5 cm.
- Construct triangle ABC using AB = 3.5 cm, BC = 3.0 cm, and AC = 5.0 cm.
- Construct triangle ABD using AB = 3.5 cm, AD = 3.8 cm, and BD = 4.5 cm.
- Join C to D to form quadrilateral ABCD.
Construction of Parallelograms
- A parallelogram has opposite sides equal and parallel.
- Diagonals of a parallelogram bisect each other.
Case 1: Two Consecutive Sides and Included Angle
- Construct a triangle with two given sides and the included angle.
- Construct another triangle using the opposite sides (equal to the given sides).
- Join the vertices to form the parallelogram.
Example: Construct parallelogram ABCD with AB = 3.0 cm, BC = 4.0 cm, and ∠B = 60°.
- Construct triangle ABC with AB = 3.0 cm, BC = 4.0 cm, and ∠ABC = 60°.
- Construct triangle ADC with DC = 3.0 cm (equal to AB) and AD = 4.0 cm (equal to BC).
- Join the vertices to form parallelogram ABCD.
Case 2: One Side and Both Diagonals
- Use half the lengths of the diagonals since they bisect each other.
- Construct a triangle with the given side and half-lengths of the diagonals.
- Extend the diagonal segments to locate the remaining vertices.
- Join the vertices to form the parallelogram.
Example: Construct parallelogram ABCD with BC = 4.5 cm, diagonal AC = 5.6 cm, and diagonal BD = 5.0 cm.
- Construct triangle OBC with OB = ½ × 5.0 cm = 2.5 cm, OC = ½ × 5.6 cm = 2.8 cm, and BC = 4.5 cm.
- Extend BO to D such that OD = OB = 2.5 cm.
- Extend CO to A such that OA = OC = 2.8 cm.
- Join AB, AD, and CD to form parallelogram ABCD.
Case 3: Two Consecutive Sides and One Diagonal
- Construct a triangle with two sides and the given diagonal.
- Construct another triangle using the opposite sides and the same diagonal.
- Join the vertices to form the parallelogram.
Example: Construct parallelogram ABCD with AB = 3.5 cm, BC = 4.5 cm, and diagonal BD = 5.5 cm.
- Construct triangle BCD with BC = 4.5 cm, CD = 3.5 cm (equal to AB), and BD = 5.5 cm.
- Construct triangle BAD with AB = 3.5 cm, AD = 4.5 cm (equal to BC), and BD = 5.5 cm.
- Join the vertices to form parallelogram ABCD.
Case 4: Two Diagonals and Included Angle
- Draw one diagonal and find its midpoint.
- Construct the given angle at the midpoint and measure half the length of the other diagonal.
- Join the vertices to form the parallelogram.
Example: Construct parallelogram ABCD with diagonal AC = 5.4 cm, diagonal BD = 4.8 cm, and angle between diagonals = 60°.
- Draw AC = 5.4 cm and find its midpoint O.
- Draw line BOD such that ∠DOC = 60°, with OB = OD = ½ × 4.8 cm = 2.4 cm.
- Join AB, BC, CD, and DA to form parallelogram ABCD.
Alternate Method for Case 4 (When Half Diagonal is Not Easily Measurable)
- If half the diagonal length (e.g., 2.25 cm) is hard to measure, start with the other diagonal.
- Draw the second diagonal and find its midpoint.
- Construct the given angle at the midpoint and measure half the length of the first diagonal.
- Join the vertices to form the parallelogram.
Example: Construct parallelogram ABCD with diagonal AC = 5.4 cm, diagonal BD = 4.5 cm, and angle between diagonals = 60°.
- Draw BD = 4.5 cm and find its midpoint O using the perpendicular bisector.
- Draw line AOC such that ∠AOD = 60°.
- From AOC, measure OA = OC = ½ × 5.4 cm = 2.7 cm.
- Join AB, BC, CD, and DA to form parallelogram ABCD.
Case 5: Two Adjacent Sides and Height
- Draw one side as the base.
- Construct a perpendicular from one end of the base and measure the given height.
- Draw a line parallel to the base through the height point.
- Use the other side's length to locate the remaining vertices on the parallel line.
- Join the vertices to form the parallelogram.
Example: Construct parallelogram ABCD with AB = 3.6 cm, BC = 4.6 cm, and height corresponding to BC = 2.6 cm.
- Draw BC = 4.6 cm.
- At C, draw CP perpendicular to BC.
- From CP, measure CE = 2.6 cm (height).
- Through E, draw QR parallel to BC.
- With B as center and radius 3.6 cm, draw an arc to intersect QR at A.
- With C as center and radius 3.6 cm (equal to AB), draw an arc to intersect QR at D.
- Join AB, AD, and CD to form parallelogram ABCD.
Construction of Trapezium
- A trapezium has one pair of parallel sides.
- Construct the trapezium using the lengths of all four sides, with one pair specified as parallel.
Example: Construct trapezium ABCD with AD parallel to BC, AD = 3.0 cm, AB = 2.5 cm, BC = 5.0 cm, and CD = 2.8 cm.
- Draw BC = 5.0 cm.
- From BC, measure BE = 3.0 cm (equal to AD).
- Construct triangle DEC with DE = 2.5 cm (equal to AB) and CD = 2.8 cm.
- With B as center and radius 2.5 cm, and D as center with radius 3.0 cm, draw arcs to intersect at A.
- Join AB and AD to form trapezium ABCD.
Construction of Rectangles
- A rectangle has opposite sides equal and all angles 90°.
- Construct by forming two right-angled triangles sharing a common side.
Case 1: Adjacent Sides
- Construct a right-angled triangle with the two given sides.
- Construct another right-angled triangle using the opposite sides.
- Join the vertices to form the rectangle.
Example: Construct rectangle ABCD with AB = 3.0 cm and BC = 5.0 cm.
- Construct triangle ABC with AB = 3.0 cm, BC = 5.0 cm, and ∠ABC = 90°.
- Construct triangle ADC with AD = 5.0 cm (equal to BC) and DC = 3.0 cm (equal to AB).
- Join the vertices to form rectangle ABCD.
Case 2: One Side and One Diagonal
- Construct a right-angled triangle with the given side and diagonal.
- Construct another right-angled triangle to complete the rectangle.
- Join the vertices to form the rectangle.
Example: Construct rectangle ABCD with BC = 4.5 cm and diagonal AC = 6.0 cm.
- Construct right-angled triangle ABC with BC = 4.5 cm, AC = 6.0 cm, and ∠ABC = 90°.
- Construct right-angled triangle ADC with DC = 4.5 cm (equal to AB) and AC = 6.0 cm.
- Join the vertices to form rectangle ABCD.
Construction of Rhombus
- A rhombus has all sides equal, and diagonals bisect each other at right angles.
- Construct using the diagonals, as they determine the vertices.
Example: Construct rhombus ABCD with diagonal AC = 6.0 cm and diagonal BD = 4.6 cm.
- Draw AC = 6.0 cm.
- Draw the perpendicular bisector of AC to find midpoint O.
- From O, measure OD = OB = ½ × 4.6 cm = 2.3 cm along the bisector.
- Join AB, BC, CD, and DA to form rhombus ABCD.
Alternate Cases for Rhombus
- If one diagonal is not easily divisible (e.g., BD = 4.6 cm gives 2.3 cm, but BD = 4.7 cm gives 2.35 cm, which is hard to measure):
- Start with the measurable diagonal and proceed similarly.
Construction of Square
- A square has equal sides and diagonals that bisect each other at right angles.
- Construct using the same method as a rhombus, as diagonals are equal.
Example: Construct a square with a diagonal of 5.4 cm (same method as rhombus).
- Draw one diagonal, say AC = 5.4 cm.
- Draw the perpendicular bisector of AC to find midpoint O.
- From O, measure OB = OD = ½ × 5.4 cm = 2.7 cm.
- Join AB, BC, CD, and DA to form the square.
Construction of Regular Hexagon
- A regular hexagon has six equal sides and each interior angle is 120°.
- Opposite sides are parallel.
- Three methods are used to construct a regular hexagon.
Method 1: Using Interior Angles
- Draw a side and construct 120° angles at both ends.
- Measure equal sides along these angles.
- Continue constructing 120° angles and sides until all six vertices are located.
- Join the vertices to form the hexagon.
Example: Construct a regular hexagon with side 3.0 cm.
- Draw AB = 3.0 cm.
- At A, draw AP such that ∠PAB = 120°; at B, draw BQ such that ∠ABQ = 120°.
- From AP, measure AF = 3.0 cm; from BQ, measure BC = 3.0 cm.
- At C, construct CD = 3.0 cm with ∠BCD = 120°.
- At F, construct FE = 3.0 cm with ∠AFE = 120°.
- Join E to D to form hexagon ABCDEF.
Method 2: Using Circumcircle
- The side of a regular hexagon equals the radius of its circumcircle.
- Draw a circle with the side length as radius.
- Mark one point on the circumference and use the same radius to mark five more points.
- Join the points to form the hexagon.
Example: Construct a regular hexagon with side 3.0 cm.
- Draw a circle with radius 3.0 cm.
- Mark point A on the circumference.
- With A as center and radius 3.0 cm, draw arcs to intersect the circle at B and F.
- With B and F as centers, draw arcs to intersect the circle at C and E.
- With C or E as center, draw an arc to intersect the circle at D.
- Join AB, BC, CD, DE, EF, and FA to form hexagon ABCDEF.
Method 3: Using Central Angle
- Each side subtends 60° at the center of the circumcircle (360° ÷ 6 = 60°).
- Construct an isosceles triangle with a 60° angle at the center.
- Draw a circle with the side length as radius.
- Mark points on the circle using the side length as radius.
- Join the points to form the hexagon.
Example: Construct a regular hexagon with side 3.0 cm.
- Draw a circle with center O and radius 3.0 cm.
- Construct isosceles triangle AOB with AB = 3.0 cm and ∠AOB = 60°.
- With B as center and radius 3.0 cm, draw an arc to intersect the circle at C.
- With C as center and radius 3.0 cm, draw an arc to intersect the circle at D.
- Continue to get points E and F.
- Join AB, BC, CD, DE, EF, and FA to form hexagon ABCDEF.
The document Chapter Notes: Construction of Polygons is a part of the Class 9 Course Mathematics Class 9 ICSE.
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FAQs on Chapter Notes: Construction of Polygons
| 1. What are the steps involved in constructing a parallelogram? |  |
Ans.To construct a parallelogram, follow these steps: 1. Draw a base line segment of a desired length. 2. At one endpoint, use a protractor to measure an angle (typically 60° or 120°) and draw a line segment of a predetermined length from that point. 3. Repeat this at the other endpoint of the base line, creating two lines. 4. Measure the distance between the ends of these two lines and ensure they are equal. 5. Connect the endpoints of the two lines to complete the parallelogram.
| 2. How can a rectangle be constructed using only a compass and a straightedge? | |
Ans.To construct a rectangle, start by drawing a base line segment (one side of the rectangle). Then, at each endpoint, use a compass to draw arcs of equal radius above and below the line. The intersection points of the arcs above and below will determine the height of the rectangle. Draw vertical lines from these intersection points to form the other two sides. Finally, connect the endpoints of the vertical lines to complete the rectangle.
| 3. What distinguishes a rhombus from other quadrilaterals in terms of construction? | |
Ans.A rhombus is a special type of quadrilateral where all four sides are of equal length. To construct a rhombus, begin by drawing a line segment for one side. Using a compass, set the radius to the length of this side and draw arcs from both endpoints to find the other two vertices. Ensure that the angles formed at the vertices are equal, thus confirming that all sides are of equal length, which characterizes a rhombus.
| 4. Can you explain the procedure for constructing a regular hexagon? | |
Ans.To construct a regular hexagon, follow these steps: 1. Start by drawing a circle with a desired radius. 2. Mark a point on the circumference as one vertex of the hexagon. 3. Using a compass set to the radius of the circle, place the compass point on this vertex and make an arc across the circumference. 4. Continue this process, marking six points around the circle, each point being the intersection of the arc with the circumference. 5. Finally, connect these points to form the hexagon.
| 5. What are the key characteristics of a trapezium that should be considered during construction? | |
Ans.A trapezium, or trapezoid, is characterized by having at least one pair of parallel sides. To construct one, first draw a line segment for the longer base. Then, from each endpoint, draw two non-parallel lines at any angle. The length of these lines will determine the height of the trapezium. Finally, draw a line segment connecting the endpoints of these two lines to complete the trapezium, ensuring that the original base remains parallel to the opposite side.
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