NCERT Solutions & Concepts for Class 7 Maths Chapter 14: Symmetry Exercise 14.1

Core Concepts: Line Symmetry and Regular Polygons

A figure has line symmetry if there is a line about which the figure can be folded so that the two parts coincide with each other. This line is called the line of symmetry or the axis of symmetry.

One of the main areas of focus in NCERT Class 7 Maths Exercise 14.1 is Regular Polygons. A polygon is said to be regular if all its sides are of equal length and all its angles are of equal measure. For example, an equilateral triangle is a regular polygon of 3 sides, and a square is a regular polygon of 4 sides.

An interesting rule to remember is that every regular polygon has as many lines of symmetry as it has sides. Therefore, a regular pentagon (5 sides) has exactly 5 lines of symmetry, and a regular hexagon (6 sides) has exactly 6 lines of symmetry. Exercise 14.1 also tests your spatial visualization through punched-hole problems, where you must identify the mirror lines that balance symmetry across two or more punched holes.

Key Definitions in Chapter 14

Line of Symmetry

The imaginary line that divides a shape or design into two identical mirror halves.

Regular Polygon

A closed geometric shape with all sides equal in length and all interior angles equal in measure.

Punched Sheet Symmetry

A visual puzzle where holes punched on a folded sheet of paper align symmetrically along the crease line when unfolded.

How to Find the Line of Symmetry for Punched Holes

1. Step 1: Identify Hole Locations — Look closely at the given shape and note down the exact spatial coordinates or relative positions of the punched holes.
2. Step 2: Find the Central Dividing Path — Draw a trial dotted line through the center of the shape (vertical, horizontal, or diagonal) that divides the geometric figure into two equal halves.
3. Step 3: Apply the Mirror Test — Ensure that if you fold the shape along your trial dotted line, each punched hole on one side falls directly on top of a corresponding punched hole on the opposite side.
4. Step 4: Confirm and Finalize — If the folded holes align perfectly, the trial line is a valid line of symmetry. Mark it clearly as a dotted line.

Number of Sides vs. Lines of Symmetry in Regular Polygons

Exam Tip: The Rectangle Diagonal Trap!

A very common mistake students make in Class 7 exams is drawing diagonal lines of symmetry for a rectangle. While a square's diagonals are lines of symmetry, a rectangle's diagonals are not.

If you cut a rectangular sheet of paper and fold it along its diagonal, the opposite corners (vertices) do not overlap perfectly. Therefore, a rectangle has only 2 lines of symmetry (horizontal and vertical lines passing through the midpoints of opposite sides), not 4!

Practice Questions with Solutions

• Q: A square sheet of paper has two symmetric punches. One punch is at the top-left corner and another is at the top-right corner. Find the line of symmetry. A: Step 1: Identify the positions of the two holes on the square paper. They are situated near the top edge, equidistant from the left and right sides. Step 2: Draw a vertical line cutting down the middle of the square. Folding along this vertical line places the top-left hole directly onto the top-right hole. Step 3: Try drawing a horizontal line across the middle. Folding along the horizontal axis does not align the holes since the bottom half contains no punches. Final answer: The square has exactly one line of symmetry, which is the vertical line passing through the midpoint of the top and bottom sides.
• Q: Determine the number of lines of symmetry for an isosceles triangle. A: Step 1: Recall that an isosceles triangle has two sides of equal length. Step 2: Identify the line dividing the angle between the equal sides. Folding along this angle bisector aligns the two equal sides perfectly. Step 3: Check if lines from the other two vertices create symmetry. Because the third side has a different length, other folds will not align the edges. Final answer: An isosceles triangle has exactly 1 line of symmetry.
• Q: A circle has two punches. One is at the top near the boundary, and the other is at the bottom near the boundary, lying on the vertical axis. How many lines of symmetry can be drawn? A: Step 1: Analyze the geometry of the circle and the punched positions. The two holes lie along the vertical diameter of the circle. Step 2: Draw a vertical line passing through both punches. Folding along this line splits each punch in half symmetrically. Step 3: Draw a horizontal line passing through the center of the circle, perpendicular to the vertical axis. Since the top and bottom punches are equidistant from the center, folding horizontally aligns the top punch directly over the bottom punch. Final answer: There are exactly 2 lines of symmetry: one vertical line passing through both holes, and one horizontal line passing through the center of the circle.
• Q: Name three shapes that have no line of symmetry. A: Step 1: Think of irregular geometric shapes whose sides and angles do not correlate symmetrically. Step 2: A scalene triangle has three unequal sides and unequal angles, meaning no fold line can divide it symmetrically. Step 3: A parallelogram has parallel opposite sides but oblique angles; folding it along any axis fails to align the vertices. Step 4: An irregular quadrilateral has no equal features, meaning no symmetry exists. Final answer: Three shapes with no line of symmetry are a scalene triangle, a parallelogram, and an irregular quadrilateral.

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