This MCQ module is based on: Constructing Squares and Rectangles
TOPIC 30 OF 41
Constructing Squares and Rectangles
🎓 Class 6
Mathematics
CBSE
Theory
Ch 8 — Playing with Constructions
⏱ ~35 min
🌐 Language: [gtranslate]
🧠 AI-Powered MCQ Assessment
▲
📐 Maths Assessment
▲
This mathematics assessment will be based on: Constructing Squares and Rectangles
Targeting Class 6 level in Geometry, with Basic difficulty.
Upload images, PDFs, or Word documents to include their content in assessment generation.
8.2 Squares and Rectangles
A square? is a four-sided figure whose sides are all equal, and all four angles are right angles (90°). A rectangle? is also four-sided with four right angles, but opposite sides are equal (length and breadth may differ).
Key Properties
Square: 4 equal sides, 4 right angles, 2 equal diagonals.Rectangle: opposite sides equal, 4 right angles, 2 equal diagonals.
Every square is a rectangle, but not every rectangle is a square.
Construct a Square of Side 6 cm
Using ruler, compass, and protractor? (or set-square):
- Draw segment PQ = 6 cm using a ruler.
- At Q, draw a line perpendicular to PQ using a protractor (90° mark).
- On this perpendicular, mark point S so that QS = 6 cm using the compass.
- At P, draw another perpendicular and mark R with PR = 6 cm.
- Join R and S. Check: RS = 6 cm, and angles at R and S are 90°.
Square PQSR of side 6 cm with right-angle marks at each vertex.
8.3 Constructing a Rectangle (Length × Breadth)
To draw a rectangle of length 6 cm and breadth 4 cm, follow the same method as the square but use two different side lengths.
- Draw PQ = 6 cm (length).
- Draw perpendicular at Q; mark S with QS = 4 cm.
- Draw perpendicular at P; mark R with PR = 4 cm.
- Join RS. Measure RS — it should equal 6 cm (opposite sides equal). Measure ∠R and ∠S — each should be 90°.
Rectangle PQSR with length 6 cm and breadth 4 cm.
A Rectangle from a Right Triangle
Given two adjacent sides AB and BC with a right angle at B, we can complete the rectangle by finding the fourth vertex using a compass (same length as opposite side). Mark out point C and B first, then use compass arcs from A (radius BC) and from C (radius AB) — the arcs meet at D.
Completing rectangle ABCD — D is found using compass arcs of appropriate radii.
Square Within a Rectangle (Construct 8.4)
Construct a rectangle of sides 8 cm and 4 cm, then place a square of side 4 cm inside it so that the centre of the square coincides with the centre of the rectangle.
Construct 8.4 — Square centred inside a rectangle.
Try This: How should the rectangle be constructed so that a diagonal divides opposite angles into two equal parts?
Answer: The rectangle must be a square — only then does a diagonal bisect the 90° corner into two 45° halves.
Answer: The rectangle must be a square — only then does a diagonal bisect the 90° corner into two 45° halves.
Figure it Out
Q. The centre of a circle is fixed at the meeting point of the two diagonals of a rectangle. Where is this centre located if a square has a "hole" (circle) centred exactly at its centre?
The centre is the intersection of the two diagonals of the square/rectangle — always at the geometric centre of the figure.
Activity: Construct Your Own Square (6 cm)
L3 ApplyMaterials: Ruler, compass, protractor, sharp pencil.
Predict: If the four sides are each 6 cm and all four angles are 90°, what will the diagonal be (roughly)?
- Use the 6-step procedure above to draw a square PQSR of side 6 cm.
- Measure the diagonal PS. Compare with \(\sqrt{2}\times 6 \approx 8.49\) cm.
- Draw both diagonals. Do they cross at the centre? Measure and check.
Both diagonals have length ≈ 8.49 cm (not an integer!), cross at the centre, and are equal. They bisect each other at right angles.
Competency-Based Questions
Scenario: A designer is laying out a rectangular notice-board 120 cm × 60 cm. She wants to mount a square poster of side 60 cm centred inside.
Q1. How much margin (width of empty space) is there on each side of the square horizontally?
L3 Apply\((120-60)/2=30\) cm on each side.
Q2. Analyse: How can she verify, using only a ruler, that the corners of the rectangle truly form right angles?
L4 AnalyseMeasure both diagonals. If they are equal, the corners are right angles (this is a property unique to rectangles).
Q3. Her apprentice claims the square in the centre creates four equal corner rectangles. Evaluate.
L5 EvaluatePartly wrong. Since the square is centred, the top and bottom strips have height 0 (square touches top and bottom of the 60 cm board). Only two strips exist: left and right, each 30 cm × 60 cm. These two are equal, but there are no "four corners".
Q4. Design a border of two nested squares (outer square side 10 cm, inner square side 6 cm) with the same centre. Describe how to construct it.
L6 Create1. Draw the 10 cm square. 2. Find its centre O by drawing the two diagonals. 3. From O, measure 3 cm (half of 6 cm) along each diagonal inward; these are approximate corners — instead, draw a 6 cm square around O using two perpendicular lines through O as axes, marking ±3 cm on each axis. 4. Join the four points.
Assertion–Reason Questions
A: Every square is a rectangle.
R: A square has four equal sides and four right angles.
R: A square has four equal sides and four right angles.
(a) — A rectangle needs 4 right angles and opposite sides equal. A square has all sides equal (a special case). R gives the property (right angles) that makes A true.
A: The diagonals of a rectangle are equal.
R: All four sides of a rectangle are equal.
R: All four sides of a rectangle are equal.
(c) — A is true. R is false; only opposite sides of a rectangle are equal.
A: In a rectangle whose diagonal divides an opposite corner into 45° + 45°, the rectangle must be a square.
R: Adjacent sides of such a rectangle must be equal.
R: Adjacent sides of such a rectangle must be equal.
(a) — Both true. Only a square has a diagonal that bisects its 90° angle, forcing the adjacent sides to be equal.
Frequently Asked Questions
What property of a square helps us construct it?
All four sides of a square are equal and every interior angle is 90 degrees. Once one side and a perpendicular are drawn accurately, the rest follows from equal measurements.
How do you draw a perpendicular at a point on a line?
Place the compass at that point, draw arcs that cut the line on both sides, then open the compass wider and draw arcs from each intersection that meet above the line. Join the meeting point to the original point to get the perpendicular.
How does constructing a rectangle differ from a square?
Both use perpendicular corners, but a rectangle uses two different side lengths (length and breadth) while a square uses the same length for all four sides.
Why do squares and rectangles always close correctly when drawn accurately?
Because opposite sides are equal and all four corners are right angles, the parallel sides align perfectly and the shape closes without error.
What real-life objects use square and rectangle constructions?
Floor tiles, book covers, door frames, window panes, graph paper, mobile screens and many geometric tile patterns are based on accurate square and rectangle constructions.
AI Tutor
Mathematics Class 6 — Ganita Prakash
Ready