Playing with Constructions[cite: 1]
Explore ruler and compass constructions, geometric patterns, circle drawing, properties and construction of squares, rectangles, diagonals, and equidistant points[cite: 1].
8.1 Artwork
Curves are any shapes that can be drawn on paper with a pencil, including straight lines, circles, and other figures. Let's explore how basic geometric tools like a ruler and compass help construct creative artwork through precision[cite: 1].
Figure 8.1: Circle Parts & Compass Placement
The fixed distance from Centre $P$ to any point on the boundary forms the circle radius[cite: 1].
Interactive Radius Simulator
Adjust the slider below to see how changing the radius $r$ scales the circle around centre $P$:
Constructing Figures[cite: 1]:
-
A Person: Combines a main head circle and multiple curved arcs drawn by placing the compass tip at strategically calculated reference points[cite: 1].
Constructed using circle arcs and centered compass points[cite: 1].
-
Wavy Wave: Drawn along a central line segment (e.g., $AB = 8\text{ cm}$) using alternating semicircles (half-circle arcs)[cite: 1].
Alternating upper and lower half-circles along central line $AB$[cite: 1].
-
Symmetrical Eyes: Constructed by intersecting two symmetrical upper and lower arcs using supporting compass points[cite: 1].
Symmetrical upper and lower arcs forming an eye shape[cite: 1].
8.2 Squares and Rectangles
Understanding the fundamental properties of rectangles and squares, their strict vertex naming conventions, and their geometric invariance under rotation.
Rectangle Properties
In a rectangle $ABCD$:
- R1: Opposite sides are equal in length ($AB = CD$ and $AD = BC$)[cite: 1].
- R2: All interior angles are equal to $90^\circ$[cite: 1].
Square Properties[cite: 1]
In a square $PQRS$[cite: 1]:
- S1: All four sides are strictly equal in length ($PQ = QR = RS = SP$)[cite: 1].
- S2: All interior angles are equal to $90^\circ$[cite: 1].
For example: $PQRS$, $QRSP$, $RSPQ$, or $SPQR$ are valid[cite: 1]. However, $PQSR$ is not a valid name because $Q$ is not directly adjacent to $S$ along the perimeter[cite: 1].
Visualizing Naming Paths
Valid Name: PQRS
Continuous path along adjacent edges[cite: 1].
Invalid Name: PQSR
Invalid: Crosses diagonally inside the shape[cite: 1].
Rotated Figures Invariance[cite: 1]
Rotated Figures: Rotating a square or rectangle does not change its side lengths or internal angles ($90^\circ$); therefore, it remains a square or rectangle regardless of its orientation[cite: 1].
Side lengths and $90^\circ$ interior angles are preserved during rotation[cite: 1].
Quick Knowledge Check
Which of the following is NOT a valid name for square $PQRS$?[cite: 1]
8.3 Constructing Squares and Rectangles
Steps to Construct a Square ($6\text{ cm}$ side):
- Step 1: Draw base line segment $PQ = 6\text{ cm}$ using a ruler.
- Step 2: Construct perpendicular lines ($90^\circ$ angles) at vertices $P$ and $Q$[cite: 1].
- Step 3: Using a compass set to $6\text{ cm}$, mark point $S$ on the perpendicular from $P$ and point $R$ on the perpendicular from $Q$[cite: 1].
- Step 4: Connect points $S$ and $R$ with a line segment to complete square $PQRS$[cite: 1].
Interactive Step-by-Step Construction Visualization
Showing Step 1: Base line segment $PQ = 6\text{ cm}$[cite: 1].
8.4 An Exploration in Rectangles[cite: 1]
For a rectangle $ABCD$ with length $AB = 7\text{ cm}$ and width $BC = 4\text{ cm}$, moving point $X$ along side $AD$ and point $Y$ along side $BC$ allows tracking of segment length $XY$ under different alignments[cite: 1]:
Key Exploration Observations[cite: 1]:
- Parallel Sub-rectangle Position: When $X$ and $Y$ are placed at equal distances from $A$ and $B$ respectively, segment $XY = AB = 7\text{ cm}$ and $ABYX$ forms a sub-rectangle[cite: 1].
- Minimum Length Segment: The minimum distance between $X$ and $Y$ occurs when $XY$ is perpendicular to parallel sides $AD$ and $BC$, yielding $XY = 4\text{ cm}$ (equal to width $BC$)[cite: 1].
- Maximum Length Segment: The maximum distance occurs between opposite vertices, where $XY$ forms a diagonal ($XY = AC = BD$)[cite: 1].
Interactive Segment Length Tracker ($XY$)
Adjust the positions of point $X$ along $AD$ and point $Y$ along $BC$ to observe how segment length $XY$ changes:
Calculated Distance $XY$ = 7.07 cm
Custom positioning along sides $AD$ and $BC$[cite: 1].
8.5 Exploring Diagonals of Rectangles and Squares
Diagonals connect opposite vertices in quadrilaterals. They reveal important geometric properties regarding segment equality and angle division in rectangles and squares[cite: 1]:
Key Properties of Diagonals[cite: 1]:
- Equality: Diagonals of a rectangle are equal in length ($PR = QS$)[cite: 1].
- Rectangle Angle Division: In a rectangle, a diagonal divides each opposite angle into two unequal parts (e.g., $30^\circ$ and $60^\circ$)[cite: 1].
- Square Angle Bisectors: In a square, adjacent sides are equal, and diagonals bisect the opposite right angles into two equal $45^\circ$ angles[cite: 1].
Rectangle vs. Square Diagonal Angle Comparison
Rectangle $PQRS$ ($PR = QS$)[cite: 1]
Diagonals divide corner right-angles into unequal angles (e.g., $30^\circ$ and $60^\circ$)[cite: 1].
Square $PQRS$ ($45^\circ - 45^\circ$ Bisectors)[cite: 1]
Diagonals bisect the $90^\circ$ right-angles into two equal $45^\circ$ angles[cite: 1].
8.6 Points Equidistant from Two Given Points[cite: 1]
When finding a point $A$ that is at an equal fixed distance ($5\text{ cm}$) from two given reference points $B$ and $C$, intersecting arcs constructed using a compass provide the exact solution[cite: 1]:
Step-by-Step Construction Procedure[cite: 1]:
- Step 1: Set compass radius to $5\text{ cm}$ and draw an arc centred at point $B$[cite: 1].
- Step 2: Keeping the compass radius fixed at $5\text{ cm}$, draw an arc centred at point $C$[cite: 1].
- Step 3: Mark the point where the two arcs intersect as point $A$ (where $AB = AC = 5\text{ cm}$)[cite: 1].
Interactive Equidistant Point Arc Construction Generator
Showing Step 1: Arc of radius $5\text{ cm}$ drawn from centre $B$[cite: 1].
Chapter Summary[cite: 1]
- All points on a circle are equidistant from its centre; this distance is the radius[cite: 1].
- A compass is used to draw circles, arcs, and measure or transfer equal lengths[cite: 1].
- Rough diagrams are helpful tools for planning geometric constructions[cite: 1].
- A rectangle can be constructed given its side lengths or one side length and a diagonal[cite: 1].
Solutions to Textbook Questions[cite: 1]
Think: Imagine marking all the points of $4\text{ cm}$ distance from point $P$. How would they look?[cite: 1]
Ans: On joining all the points, they form a circle[cite: 1].
All points at a constant $4\text{ cm}$ distance from $P$ trace out a circle[cite: 1].
Q1: What radius should be taken in the compass to get this half circle? What should be the length of $AX$?[cite: 1]
Ans: Radius = $2\text{ cm}$, $AX = 4\text{ cm}$[cite: 1].
A semicircle with radius $r = 2\text{ cm}$ has a total diameter length $AX = 4\text{ cm}$[cite: 1].
Q: Which of the following is not a name for square $PQRS$? (1. PQSR, 2. SPQR, 3. RSPQ, 4. QRSP)[cite: 1]
Ans: PQSR is not a valid name for the square[cite: 1].
Vertices must be named continuously in order (clockwise/counter-clockwise)[cite: 1]. Crossing diagonally to name $PQSR$ is invalid[cite: 1].
Q2: Identify if there are any squares in the collection[cite: 1].
Ans: Shape $A$ is a square[cite: 1].
Think: Is it possible to reason out if sides are equal and angles are right without measuring tools?[cite: 1]
Ans: Yes, the position of points on the dot grid makes it possible to determine equal sides and right angles[cite: 1].
Q1: Draw a rectangle with sides $4\text{ cm}$ and $6\text{ cm}$[cite: 1].
Ans: $\angle A = \angle B = \angle C = \angle D = 90^\circ$, $AB = CD = 4\text{ cm}$, $AD = BC = 6\text{ cm}$[cite: 1].
Q2: Draw a rectangle of sides $2\text{ cm}$ and $10\text{ cm}$[cite: 1].
Ans: $\angle P = \angle Q = \angle R = \angle S = 90^\circ$, $PQ = SR = 10\text{ cm}$, $PS = QR = 2\text{ cm}$[cite: 1].
Q3: Is it possible to construct a 4-sided figure in which all angles are $90^\circ$ but opposite sides are not equal?[cite: 1]
Ans: No[cite: 1].
| Distance of X from A | Distance of Y from B | Length of XY |
|---|---|---|
| $5\text{ mm}$ | $3\text{ cm}$ | $7.4\text{ cm}$ |
| $1\text{ cm}$ | $1\text{ cm}$ | $7\text{ cm}$ |
| $2\text{ cm}$ | $4\text{ cm}$ | $7.3\text{ cm}$ |
| $5\text{ mm}$ | $5\text{ mm}$ | $7\text{ cm}$ |
| $1\text{ cm}$ | $1\text{ cm}$ | $7\text{ cm}$ |
| $1\text{ cm } 5\text{ mm}$ | $1\text{ cm } 5\text{ mm}$ | $7\text{ cm}$ |
Q: How does length $XY$ compare to $AB$? What is the shape of $ABYX$?[cite: 1]
Ans: (i) $XY = AB$, (ii) $ABYX$ is a rectangle[cite: 1].
Q: How does the farthest distance between $X$ and $Y$ compare with $AC$ / $BD$?[cite: 1]
Ans: The farthest distance between $X$ and $Y$ is equal to $AC$ or $BD$[cite: 1].
Q: Give side lengths of rectangles that cannot be divided into 2 or 3 identical squares[cite: 1]:
- Cannot be divided into 2 identical squares: Length = $4\text{ cm}$, Breadth = $2.5\text{ cm}$[cite: 1].
- Cannot be divided into 3 identical squares: Length = $7\text{ cm}$, Breadth = $2\text{ cm}$[cite: 1].
Q4 (Square with a Hole): Where should the centre of the circular hole be?[cite: 1]
Ans: At the meeting point of the two line segments connecting opposite vertices (diagonals)[cite: 1].
The centre of symmetry lies at the intersection of diagonals[cite: 1].
Q1: How should a rectangle be constructed so that the diagonal divides opposite angles into equal parts?[cite: 1]
Ans: Two adjacent sides must be equal, making the rectangle a square[cite: 1].
Q2: Construct a rectangle in which a diagonal divides opposite angles into $45^\circ$ and $45^\circ$. What do you observe about the sides?[cite: 1]
Ans: All four sides are equal (it forms a square)[cite: 1].
Interactive Diagonal Angle Bisector Explorer
Adjust the height slider to see how equalizing adjacent sides transforms unequal diagonal angles into equal $45^\circ$ bisectors[cite: 1]:
Diagonal Angle Division at Vertex A: 26.6° and 63.4°
Adjacent sides are unequal ($BC \neq AB$). Diagonal divides corner angle into unequal parts[cite: 1].