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📖 Mathematics

Std 6
Books Book Guide
1 2 3 4 5 6 7 8 9 10
Chapters
1 PATTERNS IN MATHEMATICS 2 Lines and Angles 3 Number Play 4 Data Handling and Presentation 5 Idli-Vada Game PRIME TIME 6 PERIMETER AND AREA 7 Fractions 8 Playing with Constructions 9 SYMMETRY 10 Integers (The Other Side of Zero)
8
Chapter 8
Skill: 30%

Playing with Constructions

Playing with Constructions

Chapter at a Glance

This chapter covers geometric constructions using a ruler and compass. Students explore the properties of circles, squares, and rectangles, and learn how to construct them step-by-step. The chapter guides students through drawing geometric artwork (like waves and eyes), partitioning rectangles into identical squares, analyzing the properties of diagonals, locating points equidistant from two targets, and constructing composite shapes (like a symmetric house with a curved arch).

Key Definitions & Terminology

  • Compass: A geometric instrument with a pivot point and a pencil holder, used to draw circles, arcs, and to transfer distances.
  • Ruler: A straight-edge tool with measurement markings, used to draw straight lines and measure lengths.
  • Curve: Any shape drawn on paper, including straight lines, circles, and irregular curved lines.
  • Circle: A closed curve where all points are at an equal distance (radius) from a fixed center.
  • Radius: The distance from the center of a circle to any point on its boundary.
  • Rectangle: A quadrilateral with opposite sides of equal length and all interior angles measuring $90^\circ$.
  • Square: A regular quadrilateral with all four sides equal in length and all interior angles measuring $90^\circ$.
  • Rhombus: A quadrilateral with all four sides equal in length, but whose interior angles are not necessarily $90^\circ$ (a non-square equilateral quadrilateral).
  • Diagonal: A line segment connecting two opposite corners of a polygon.
  • Perpendicular Lines: Two intersecting lines that meet at a right angle ($90^\circ$).

Formulas, Rules & Properties

  • Quadrilateral Naming Rule:
    A valid name for a square or rectangle must list the corner vertices in consecutive circular order around the boundary (e.g. $ABCD$ or $ADCB$, but not $ACBD$ or $ABDC$).
  • Rotational Invariance:
    Rotating a square or rectangle in 2D space does not change its side lengths or angles; it remains a square or rectangle.
  • Rectangle Diagonal Properties:
  • The two diagonals are equal in length ($AC = BD$).
  • In general, diagonals of a rectangle do not bisect its corner angles unless the rectangle is a square.
  • Square Diagonal Properties:
  • The two diagonals are equal in length and perpendicular to each other.
  • The diagonals bisect the opposite $90^\circ$ angles into two equal $45^\circ$ angles.
  • Rectangle Partitioning:
    A rectangle can be divided into $N$ identical squares of side $x$ if and only if its dimensions are $Nx$ and $x$ (aspect ratio $N:1$).
  • Locus of Equidistant Points:
  • The set of all points at distance $d$ from a center point $P$ forms a circle of radius $d$ centered at $P$.
  • A point equidistant from two vertices $B$ and $C$ at a distance $r$ is located at the intersection of two arcs of radius $r$ centered at $B$ and $C$.

Core Concepts & Topics

  • Compass for Distance Transfer:
    Without using ruler numbers, a distance can be transferred by placing the compass tips on points $A$ and $B$, then moving the compass to pivot on a new point and drawing a marking arc.
  • Dot Grid Reasoning:
    Checking if sides are equal or angles are perpendicular on dot paper by analyzing grid vector steps (e.g. a vertical step of 3 and horizontal step of 4 vs. a horizontal step of 3 and vertical step of 4).
  • Moving Points along Rectangle Boundaries:
  • Consider a rectangle $ABCD$ where point $X$ moves along side $AD$ and point $Y$ moves along side $BC$:
    • The minimum distance between $X$ and $Y$ is equal to side $AB$. This occurs when $AX = BY$ (making $ABYX$ a rectangle).
    • The maximum distance between $X$ and $Y$ occurs when they are at opposite vertices (e.g. $X=A$ and $Y=C$), which is equal to the diagonal length $AC$.
  • Symmetric Construction Artwork:
  • Wavy Wave: Alternating semicircles drawn along a straight line segment.
  • Eyes: Formed by the symmetrical intersection of two opposite circular arcs.
  • Symmetric House: Composed of a square base, an equilateral triangle roof, and a downward-bulging semicircular arch ceiling connecting the two roof corners.
  • Square with a Hole: A square with a concentric circular hole. The center of the circle is located at the intersection of the square's two diagonals.

Worked Examples

  • Wavy Wave Semicircle Radius:
  • Question: A wavy wave is drawn along segment $AB = 8\text{ cm}$. The first wave is a semicircle on $AX$, where $X$ is the midpoint of $AB$. What radius and $AX$ length are needed?
    Solution: The segment $AX$ has length $8 / 2 = 4\text{ cm}$. Since $AX$ acts as the diameter of the semicircle, the radius is $4 / 2 = \mathbf{2\text{ cm}}$, and $AX = \mathbf{4\text{ cm}}$.
  • Valid Naming Identification:
  • Question: Which of the following is not a valid name for a square with corners labeled $P, Q, S, R$ in clockwise order? (1) $PQSR$, (2) $SPQR$, (3) $RSPQ$, (4) $QRSP$.
    Solution: Moving consecutively along the sides gives $P \rightarrow Q \rightarrow S \rightarrow R \rightarrow P$. Therefore, $PQSR$ is a valid name. The options listed in the source textbook are $PQSR$ vs $SPQR$. Wait, if corners are $P, Q, S, R$ in clockwise order, then consecutive corners are $P, Q, S, R$. The name $PQSR$ is valid. Let's see: if consecutive corners are $P, Q, R, S$, then $PQSR$ is invalid. The textbook answers that $PQSR$ is not a name for the square because the consecutive corners are $P, Q, R, S$ (with $S$ and $R$ swapped in the drawing).
  • Partitioning Rectangles:
  • Question: Give dimensions of a rectangle that cannot be divided into (a) two identical squares, (b) three identical squares.
    Solution:
    • Cannot make 2 squares: e.g. a rectangle of $4\text{ cm} \times 2.5\text{ cm}$ (aspect ratio $1.6 \neq 2$).
    • Cannot make 3 squares: e.g. a rectangle of $7\text{ cm} \times 2\text{ cm}$ (aspect ratio $3.5 \neq 3$).
  • Moving Points on a Rectangle:
  • Question: In a rectangle $ABCD$ ($AB = 7\text{ cm}, BC = 4\text{ cm}$), find the length of $XY$ when:
    • $X$ is $5\text{ mm}$ from $A$, $Y$ is $3\text{ cm}$ from $B \rightarrow XY \approx \mathbf{7.4\text{ cm}}$.
    • $X$ is $1\text{ cm}$ from $A$, $Y$ is $1\text{ cm}$ from $B \rightarrow XY = AB = \mathbf{7\text{ cm}}$.
    • $X$ is $2\text{ cm}$ from $A$, $Y$ is $4\text{ cm}$ from $B \rightarrow XY \approx \mathbf{7.3\text{ cm}}$.
    • In general, when $AX = BY$, the length of $XY = AB = \mathbf{7\text{ cm}}$, and the figure $ABYX$ is a rectangle.
  • Side and Diagonal Construction:
  • Question: Construct a rectangle $ABCD$ with base $CD = 5\text{ cm}$ and diagonal $BD = 7\text{ cm}$.
    Solution:
    • Step 1: Draw horizontal line segment $CD = 5\text{ cm}$.
    • Step 2: Draw a perpendicular line $l$ extending upwards from vertex $C$.
    • Step 3: Set the compass radius to $7\text{ cm}$. Place the pivot on $D$ and draw an arc that intersects line $l$. Label this intersection point $B$.
    • Step 4: Draw a perpendicular to $CD$ at $D$, and a perpendicular to $CB$ at $B$. Their intersection point is $A$. Complete the rectangle.
  • Angle and Diagonal Construction:
  • Question: Construct a rectangle in which a diagonal divides the corner angle into $60^\circ$ and $30^\circ$.
    Solution: Draw a base segment $AB$. At $A$, draw a line at an angle of $60^\circ$ to $AB$. At $B$, construct a perpendicular to $AB$. The intersection of these two lines is vertex $C$. The diagonal is $AC$. Construct perpendiculars from $A$ and $C$ to locate the final vertex $D$.
  • Equilateral Quadrilateral Construction:
  • Question: Can a quadrilateral have all sides equal in length but not be a square?
    Solution: Yes, a rhombus. To construct it: draw a base line $BC = 5\text{ cm}$. Draw an arc of radius $5\text{ cm}$ centered at $B$ at an angle other than $90^\circ$. Draw another arc of radius $5\text{ cm}$ centered at $C$. The intersection of these arcs locates the remaining vertices, forming an oblique equilateral diamond.

Practical Activities & Experiments

  • Compass Familiarization Craft: Drawing concentric circles and overlapping flower patterns on paper.
  • Dot Grid Symmetrical Square Layout: Sketching a central rectangle and placing four smaller squares at identical diagonal offsets.
  • Rectangle Creasing/Folding Experiment: Creasing a rectangular sheet along both diagonals to verify that they are equal in length.
  • House Model Construction:
  • Constructing a base square $BCED$ of side $5\text{ cm}$.
  • Placing the compass pivot on $B$ and $C$ with a radius of $5\text{ cm}$ to draw intersecting arcs that locate the roof vertex $A$ (forming equilateral triangle $\triangle ABC$).
  • Placing the compass pivot on $A$ with a radius of $5\text{ cm}$ to draw the curved arch ceiling from $B$ to $C$.
  • Centering a Hole: Drawing the diagonals of a paper square, using their intersection point as the center, and drawing a circle to create a centered hole.
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