Constructions and Tilings — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Constructions and Tilings" — 6 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Constructions and Tilings" — 6 important questions with detailed answers for CBSE b…
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NCERT Solutions for Class 7 Mathematics — Constructions and Tilings. Step-by-step answers to all 6 textbook questions from this chapter, written for CBSE board preparation and free to use.
Constructions and Tilings — All 6 Questions Solved
Q1. What are the basic tools used in geometric constructions? List them and explain one tool's use.
Basic Tools for Geometric Constructions:
1. Ruler (Straight Edge)
Used to: Draw straight lines and measure lengths
2. Compass
Used to: Draw circles and arcs of specific radii
3. Set Square / Triangle (30-60-90 or 45-45-90)
Used to: Draw perpendicular lines and angles of 30°, 45°, 60°, 90°
4. Protractor
Used to: Measure and draw angles in degrees
5. Divider
Used to: Transfer distances and mark equal segments on lines
6. Pencil
Used to: Mark points and draw construction lines
---
Detailed Explanation: Compass
The compass is an essential tool for geometric constructions.
Parts of a Compass:
• Two pointed ends
• One end holds a sharp needle (center point)
• Other end holds a pencil or leads for drawing
• Adjustable legs joined at a pivot
Uses of Compass:
1. Drawing circles and arcs
2. Marking equal distances (transferring lengths)
3. Creating perpendicular bisectors
4. Drawing angle bisectors
5. Constructing congruent segments
6. Creating perpendicular and parallel lines
Example Use:
To draw a circle with center O and radius 3 cm:
• Place the compass needle at point O
• Open the compass to a width of 3 cm using the ruler
• Rotate the compass around point O to draw the circle
• All points on the circle are exactly 3 cm from O
Precision:
The compass allows us to create exact geometric shapes based on precise measurements, which is crucial for accurate constructions.
Q2. Construct a perpendicular to a line AB at a point P on the line.
Constructing a Perpendicular to Line AB at Point P:
---
Steps:
Step 1: Draw the base line
Draw a straight line AB with point P marked on it.
A ____________P____________ B
Step 2: Mark equal arcs on either side of P
Using a compass with any suitable radius:
• Place the compass needle at P
• Draw arcs on both sides of P on line AB
• Mark these intersection points as X and Y
• Now PX = PY (equidistant from P)
A ___X___P___Y___ B
Step 3: Draw arcs above and below the line
Using a compass with a radius greater than PX:
• Place the compass needle at X
• Draw an arc above the line AB
• Using the SAME radius, place the compass needle at Y
• Draw an arc below the line AB (or above)
• Mark the intersection point of these arcs as C
Example (arc above):
C
/|\
/ | \
/ | \
/ | \
A_X_P_Y_B
Step 4: Draw the perpendicular
Draw a straight line through P and C using a ruler.
Line CP is perpendicular to line AB.
Angle CPA = Angle CPB = 90°
---
Why This Works:
Because X and Y are equidistant from P, and C is equidistant from X and Y, point C lies on the perpendicular bisector of XY, which passes through P. This perpendicular bisects the line segment XY at P.
---
Verification:
You can verify using a protractor:
• Measure angle CPA = 90°
• Measure angle CPB = 90°
Q3. Define line symmetry and rotational symmetry. Give examples of each.
Line Symmetry (Reflection Symmetry):
Definition: A shape has line symmetry if it can be folded along a line so that both halves match exactly.
The line is called the line of symmetry or axis of symmetry.
Properties:
• The line divides the shape into two identical halves
• Each point on one side has a corresponding point on the other side
• The corresponding points are equidistant from the line of symmetry
Examples of Line Symmetry:
1. Isosceles triangle: 1 line of symmetry (vertical)
2. Rectangle: 2 lines of symmetry (horizontal and vertical)
3. Square: 4 lines of symmetry (2 diagonals + 1 horizontal + 1 vertical)
4. Circle: Infinite lines of symmetry (any diameter)
5. Equilateral triangle: 3 lines of symmetry
6. Butterfly: 1 line of symmetry (vertical)
7. Letter A: 1 line of symmetry (vertical)
---
Rotational Symmetry:
Definition: A shape has rotational symmetry if it looks the same after rotating it less than 360°.
The center of rotation is called the center of symmetry.
Order of Rotational Symmetry: The number of times a shape coincides with itself during one complete rotation (360°).
Example:
If a shape rotates 90° and looks the same, it has rotational symmetry of order 4.
(Because 360° ÷ 90° = 4)
Examples of Rotational Symmetry:
1. Square: Rotational symmetry of order 4 (90°, 180°, 270°, 360°)
2. Equilateral triangle: Order 3 (120°, 240°, 360°)
3. Rectangle: Order 2 (180°, 360°)
4. Star (5-pointed): Order 5 (72°)
5. Circle: Infinite rotational symmetry
6. Regular pentagon: Order 5 (72°)
7. Letter S: Order 2 (180°)
---
Comparison Table:
| Property | Line Symmetry | Rotational Symmetry |
|----------|---------------|---------------------|
| Type | Reflection | Rotation |
| Action | Fold or mirror | Turn around center |
| Angle | Fold along a line | Rotate around a point |
| Square | 4 lines | Order 4 |
| Circle | Infinite lines | Infinite order |
Q4. What are tilings (tessellations)? How do regular triangles tile a plane? Show the arrangement.
Tilings (Tessellations):
Definition: A tiling or tessellation is a pattern of shapes that completely covers a plane without gaps or overlaps.
The shapes fit together perfectly to create a continuous pattern.
Requirements for Tiling:
1. The shapes must cover the entire plane
2. There should be no gaps between shapes
3. There should be no overlapping shapes
4. The pattern can repeat infinitely
---
Regular Tilings:
Only three regular polygons can tile a plane alone:
1. Equilateral triangles
2. Squares
3. Regular hexagons
---
How Regular Triangles Tile a Plane:
Condition Check:
Each interior angle of an equilateral triangle = 60°
At each vertex in the tiling: 360° ÷ 60° = 6 triangles
So exactly 6 equilateral triangles meet at each vertex.
Since 60° divides 360° evenly, triangles CAN tile a plane. ✓
---
Arrangement Pattern:
How to arrange:
1. Start with one equilateral triangle
2. Place another triangle sharing one complete side
3. Continue placing triangles to form rows
4. Each new triangle shares at least one side with existing triangles
5. At each vertex, exactly 6 triangles meet
Visual Representation:
/\
/ \
/____\
/\ /\
/ \ / \
\____/\____/
/\ /\
/ \ / \
\____/\____/
The pattern shows:
• Triangles pointing up: △
• Triangles pointing down: ▽
• They interlock perfectly
• At each vertex: 6 triangles meet
• No gaps or overlaps
---
Angle Verification:
At each vertex in the triangle tiling:
6 × 60° = 360° ✓
This confirms the tiling is possible.
---
Other Examples of Tilings:
1. Square tiling: 4 squares per vertex (90° × 4 = 360°)
2. Hexagonal tiling: 3 hexagons per vertex (120° × 3 = 360°)
3. Mixed tilings: Squares and triangles together
4. Regular hexagons: 3 hexagons per vertex (120° × 3 = 360°)
Applications:
• Honeycomb structures (hexagonal)
• Tile flooring (square or hexagonal)
• Islamic geometric patterns
• Wallpapers and decorative designs
Q5. Construct an angle of 60° using compass and ruler. Explain why this construction works.
Constructing a 60° Angle using Compass and Ruler:
---
Steps of Construction:
Step 1: Draw the base ray
Draw a ray OA starting from point O.
O_____________A
Step 2: Mark an arc from point O
Using a compass with any suitable radius (say 3 cm):
• Place the compass needle at O
• Draw an arc intersecting ray OA
• Mark the intersection point as B
O___B___________A
Step 3: Draw an arc from point B
Using the SAME radius as Step 2:
• Place the compass needle at B
• Draw an arc intersecting the arc from Step 2
• Mark the intersection point as C
C
*
\ (arc)
\
\ (arc)
O___B___A
Step 4: Draw the angle line
Draw a straight line from O through C using a ruler.
Angle BOC = 60°
C
/|
/ |
/ |
/60°|
O___B
Final Construction:
C
/
/ 60°
/
/
O____B____A
---
Why This Construction Works:
Explanation:
1. OB = OC (both equal to the radius used in compass) — [Both are radii]
2. BC = OB = OC (we used the same radius for the arc from B)
Triangle Analysis:
Triangle OBC has:
• OB = BC = OC (all three sides equal)
• This makes triangle OBC an EQUILATERAL TRIANGLE
Property of Equilateral Triangle:
• All angles in an equilateral triangle are equal
• Sum of angles in any triangle = 180°
• Each angle = 180° ÷ 3 = 60°
• Therefore, angle BOC = 60° ✓
---
Mathematical Proof:
Given:
• Radius from O = r
• Arc from B with same radius r intersects the arc from O at C
Then:
• OB = r (radius from O)
• OC = r (radius from O)
• BC = r (since C lies on arc of radius r from B)
Since OB = OC = BC, triangle OBC is equilateral.
In an equilateral triangle, all angles = 60°.
Therefore, angle AOC = 60° ✓
---
Verification:
You can verify this using a protractor:
• Place protractor at O
• Align baseline with OA
• Read the angle where OC intersects = 60°
Q6. Identify and name all lines of symmetry in a square ABCD.
Identifying Lines of Symmetry in a Square ABCD:
---
Square Setup:
Let the square have vertices:
A _______ B
| |
| |
| |
D _______ C
---
Total Lines of Symmetry: A square has 4 lines of symmetry
---
Line 1: Horizontal Line of Symmetry
Line through the midpoint of AB and midpoint of DC
A ___M1__ B
| | |
|____|____|
D ___M2__ C
Line: M1M2 (horizontal midline)
This line folds the square so that:
• Top half (A-B-M1-M2) matches with bottom half (M1-M2-C-D)
---
Line 2: Vertical Line of Symmetry
Line through the midpoint of AD and midpoint of BC
A | B
M4 | | M3
| |
D C
Line: M4M3 (vertical midline)
This line folds the square so that:
• Left half (A-D-M4-M3) matches with right half (M3-M4-B-C)
---
Line 3: Diagonal from A to C
A \ B
| \ |
| \ |
| \ |
D ____\ C
Line: AC (diagonal from top-left to bottom-right)
This line folds the square so that:
• Triangle ABC matches with triangle ADC
• Vertex B maps to vertex D
---
Line 4: Diagonal from B to D
A / B
| / |
| / |
| / |
D /_____ C
Line: BD (diagonal from top-right to bottom-left)
This line folds the square so that:
• Triangle ABD matches with triangle CBD
• Vertex A maps to vertex C
---
Summary Table:
| Line Number | Type | Description | Folds |
|-------------|------|-------------|-------|
| 1 | Horizontal | Through midpoints of AB and DC | Top ↔ Bottom |
| 2 | Vertical | Through midpoints of AD and BC | Left ↔ Right |
| 3 | Diagonal | From vertex A to vertex C | Triangle ABC ↔ Triangle ADC |
| 4 | Diagonal | From vertex B to vertex D | Triangle ABD ↔ Triangle CBD |
---
Verification:
For each line of symmetry:
• Every point on one side has a corresponding equidistant point on the other side ✓
• When folded along the line, both halves match exactly ✓
• The line divides the square into two congruent shapes ✓
---
Conclusion:
A square has exactly 4 lines of symmetry:
• 2 lines through the midpoints of opposite sides (horizontal and vertical)
• 2 lines along the diagonals
Constructions and Tilings — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. Which geometric instrument is primarily used to draw circles and arcs?
- A. Ruler
- B. Protractor
- C. Compass
- D. Set Square
Answer: C. Compass — A compass is specifically designed to draw circles and arcs of various radii.
MCQ 2. What is the first step when preparing to construct a perpendicular bisector of a line segment AB?
- A. Draw a line passing through the midpoint of AB
- B. Place the compass at point A and draw an arc
- C. Measure the length of AB with a ruler
- D. Draw a line segment AB
Answer: D. Draw a line segment AB — The very first step in constructing a perpendicular bisector is to have the line segment AB already drawn.
MCQ 3. What is the term for covering a surface with geometric shapes without any gaps or overlaps?
- A. Symmetry
- B. Reflection
- C. Tessellation
- D. Rotation
Answer: C. Tessellation — Tessellation, also known as tiling, is the process of covering a plane using one or more geometric shapes without any overlaps or gaps.
MCQ 4. Which of the following regular polygons can always tessellate a plane by itself?
- A. Regular Pentagon
- B. Regular Octagon
- C. Equilateral Triangle
- D. Regular Heptagon
Answer: C. Equilateral Triangle — An equilateral triangle has an interior angle of 60 degrees. Since 360 is divisible by 60 (360/60 = 6), equilateral triangles can tessellate.
MCQ 5. To construct an angle of 90 degrees using a compass and ruler, which angle construction is typically done as an intermediate step or is inherent in the method?
- A. 30 degrees
- B. 45 degrees
- C. 60 degrees
- D. 120 degrees
Answer: C. 60 degrees — A common method to construct 90 degrees involves constructing a 60-degree angle and its supplementary angle of 120 degrees, then bisecting the angle between them or constructing perpendiculars based on arcs derived from 60 and 120 degree points.
MCQ 6. What does an angle bisector do to an angle?
- A. Makes it twice as large
- B. Divides it into two equal parts
- C. Rotates it by 90 degrees
- D. Measures its size
Answer: B. Divides it into two equal parts — An angle bisector is a ray that divides an angle into two angles of equal measure.
MCQ 7. What must be the sum of the interior angles of the polygons meeting at a single vertex in a tessellation?
- A. 90 degrees
- B. 180 degrees
- C. 270 degrees
- D. 360 degrees
Answer: D. 360 degrees — For shapes to tessellate, the sum of the angles around any common vertex must be exactly 360 degrees, with no gaps or overlaps.
MCQ 8. If you construct the perpendicular bisector of a line segment AB, what is true about any point on the bisector?
- A. It is equidistant from points A and B
- B. It is equidistant from the endpoints of any other line segment
- C. It forms an acute angle with AB
- D. It is the midpoint of AB
Answer: A. It is equidistant from points A and B — Any point on the perpendicular bisector of a line segment is equidistant from the two endpoints of the segment.
MCQ 9. Which of the following regular polygons *cannot* tessellate a plane by itself?
- A. Square
- B. Regular Hexagon
- C. Equilateral Triangle
- D. Regular Pentagon
Answer: D. Regular Pentagon — A regular pentagon has an interior angle of 108 degrees. Since 360 is not divisible by 108, regular pentagons cannot tessellate a plane by themselves.
MCQ 10. To construct a line parallel to a given line 'l' through a point 'P' not on 'l', which geometric property is most commonly used?
- A. Vertically opposite angles are equal
- B. Corresponding angles are equal
- C. Angles on a straight line sum to 180 degrees
- D. Sum of angles in a triangle is 180 degrees
Answer: B. Corresponding angles are equal — The most common construction method for parallel lines through a point uses the property that if a transversal intersects two lines such that corresponding angles (or alternate interior angles) are equal, then the lines are parallel.
MCQ 11. Which tool is used to measure and draw angles of specific measures?
- A. Ruler
- B. Compass
- C. Protractor
- D. Divider
Answer: C. Protractor — A protractor is a flat, semicircular tool used specifically for measuring and drawing angles.
MCQ 12. In a tessellation made of regular hexagons, how many hexagons meet at each vertex?
- A. Two
- B. Three
- C. Four
- D. Five
Answer: B. Three — Each interior angle of a regular hexagon is 120 degrees. Since 360 / 120 = 3, exactly three regular hexagons meet at each vertex in a regular tessellation.
MCQ 13. What is the correct order of steps to construct an angle bisector of angle ABC?
- A. 1. Place the compass at B and draw an arc intersecting BA and BC. 2. Draw a ray from B through the intersection point of the arcs. 3. Place the compass at the intersection points on BA and BC, draw arcs to intersect each other.
- B. 2, 1, 3
- C. 1, 3, 2
- D. 3, 1, 2
Answer: C. 1, 3, 2 — First, draw an arc from the vertex B. Then, from where that arc intersects the angle's arms (BA and BC), draw two more arcs that intersect each other. Finally, draw a ray from B through that intersection point.
MCQ 14. Why do squares tessellate a plane?
- A. Because they have four equal sides
- B. Because their interior angle is 90 degrees, which divides 360 degrees evenly
- C. Because they are symmetrical
- D. Because they can be stacked easily
Answer: B. Because their interior angle is 90 degrees, which divides 360 degrees evenly — A square has an interior angle of 90 degrees. Since 360 / 90 = 4, exactly four squares can meet at a vertex without any gaps or overlaps, allowing them to tessellate.
MCQ 15. To construct an angle of 45 degrees, which two fundamental constructions are typically combined?
- A. Copying a segment and bisecting an angle
- B. Constructing a perpendicular and bisecting an angle
- C. Constructing a 60-degree angle and copying an angle
- D. Bisecting a segment and constructing a perpendicular
Answer: B. Constructing a perpendicular and bisecting an angle — To construct 45 degrees, you first construct a 90-degree angle (which involves constructing a perpendicular to a line), and then you bisect this 90-degree angle.
MCQ 16. A straightedge (ruler without markings) is primarily used for:
- A. Measuring lengths
- B. Drawing straight lines
- C. Drawing circles
- D. Drawing specific angles
Answer: B. Drawing straight lines — A straightedge is used for drawing straight lines or line segments without any measurement.
MCQ 17. Which of these is an example of tiling seen in nature?
- A. Rainbow
- B. Clouds
- C. Honeycomb
- D. Ocean Waves
Answer: C. Honeycomb — Honeycombs are a classic natural example of tessellation, formed by regular hexagonal cells.
MCQ 18. When constructing a perpendicular to a line from a point *outside* the line, how many arcs typically need to be drawn from the external point to intersect the line?
- A. One
- B. Two
- C. Three
- D. Four
Answer: A. One — To construct a perpendicular from an external point, one arc is drawn from the external point to intersect the line at two distinct points. Subsequent arcs are drawn from these two new points.
MCQ 19. What is a 'semi-regular tessellation'?
- A. A tessellation with gaps and overlaps
- B. A tessellation using only one type of regular polygon
- C. A tessellation using two or more types of regular polygons
- D. A tessellation using irregular polygons only
Answer: C. A tessellation using two or more types of regular polygons — A semi-regular tessellation is a tessellation of the plane by two or more types of regular polygons, where the arrangement of polygons is the same at every vertex.
MCQ 20. If you construct a 60-degree angle, then bisect it, and then bisect one of the newly formed angles, what is the smallest angle you can obtain?
- A. 15 degrees
- B. 20 degrees
- C. 22.5 degrees
- D. 30 degrees
Answer: A. 15 degrees — Starting with a 60-degree angle, bisecting it gives 30 degrees. Bisecting the 30-degree angle then gives 15 degrees.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.