Introduction to Similar Figures
Similar figures, congruency of shapes and dilation.
Exploration of Geometric Figures – Introduction (Class 8 Maths)
Chapter 8 of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) takes students on an exciting journey into the world of geometric shapes — examining how figures relate to each other through congruence, similarity, and transformations. Rather than memorising isolated facts, this chapter builds visual thinking and spatial reasoning skills that are essential for Class 9 and Class 10 geometry.
The introduction establishes three core ideas: what makes two figures congruent (exactly the same shape and size), what makes them similar (same shape but different size), and how figures can be transformed — flipped, rotated, or stretched — while preserving these relationships. Understanding these ideas early makes topics like triangle congruence, similarity theorems, and coordinate geometry far easier to master later.
What Are Congruent Figures?
Think of two flowers of identical species picked from the same plant — they look the same and measure the same. In mathematics, two triangles, circles, squares, or any polygons can be congruent in exactly this way. One figure can be placed exactly on top of the other so that every point matches.
Congruence applies to all kinds of geometric figures — line segments, angles, triangles, quadrilaterals, and circles. The key requirement is both same shape and same size — just one of these is not enough.
Congruence of Line Segments
Two line segments are congruent if and only if their lengths are equal. If AB and PQ are two line segments with AB = PQ, then we write AB ≅ PQ. You can verify this by placing one segment over the other — they will overlap completely.
If AB = PQ (in length), then AB ≅ PQ
Congruence of Angles
Two angles are congruent when their measures (in degrees) are equal. If ∠ABC = ∠PQR = 40°, then we write ∠ABC ≅ ∠PQR. You can check this by placing one angle over the other — the rays and vertex will coincide perfectly.
If ∠ABC = ∠PQR (in degrees), then ∠ABC ≅ ∠PQR
Congruence of Circles
For two circles to be congruent, their radii must be equal. Two circles with the same radius are identical in every way — one can be perfectly placed over the other.
Circle₁ ≅ Circle₂ ⟺ r₁ = r₂
Congruence of Squares
Two squares are congruent when their sides are equal. Since all angles in a square are already 90°, the side length alone determines congruence. Equivalently, two squares whose diagonals are equal are also congruent.
Square₁ ≅ Square₂ ⟺ side₁ = side₂ (or diagonal₁ = diagonal₂)
Congruence of Triangles — Corresponding Parts
Triangles are the most studied case of congruence. Two triangles are congruent when all three pairs of corresponding sides are equal and all three pairs of corresponding angles are equal. This is written as △ABC ≅ △PQR.
| Corresponding Sides | Corresponding Angles |
|---|---|
| AB = PQ | ∠A = ∠P |
| BC = QR | ∠B = ∠Q |
| AC = PR | ∠C = ∠R |
Transformations — Flip, Slide, and Rotation
A transformation is a way of moving or changing a figure in a plane without altering its essential properties. There are three basic transformations you need to know for Class 8: Flip (Reflection), Slide (Translation), and Rotation. Importantly, congruent figures remain congruent through all of these transformations.
A Flip is a transformation in which a plane figure is reflected across a line — called the line of reflection or mirror line. The result is a mirror image of the original figure. The reflected figure is congruent to the original: every point in the image is at the same distance from the mirror line as the corresponding point in the original, but on the opposite side.
Real-life examples of flips include a pair of slippers (left and right are mirror images), the reflection of a tree in a lake, or the wings of a butterfly. In coordinate geometry (Class 10), flips are described using rules like (x, y) → (−x, y) for reflection across the y-axis.
A Rotation turns a figure around a fixed point called the centre of rotation. The key property is that every point on the figure traces a circular arc around the centre, and the distance from the centre to any point on the shape stays exactly the same throughout the rotation.
- Centre of rotation — the fixed point around which the figure turns (it does not move).
- Angle of rotation — the angle through which the figure is turned (e.g. 90°, 180°, 270°).
- Direction — clockwise or anticlockwise.
A key insight: if two shapes are congruent, they remain congruent even after one of them is moved (slid), rotated, or flipped. The transformation does not destroy congruence.
Similar Figures — Same Shape, Different Size
Think of a photograph enlarged on a photocopier — the enlarged photo and the original look identical in shape, but one is bigger. In mathematics, two butterflies of the same species but different sizes, or two dogs of the same breed at different ages, are everyday illustrations of similar figures.
How to Check if Two Polygons Are Similar
- Step 1: Check that all pairs of corresponding angles are equal.
- Step 2: Check that the ratios of all pairs of corresponding sides are equal (proportional).
- If both conditions hold, the figures are similar.
Worked Example: Similar Rectangles
Consider a rectangle with sides 3 cm and 2 cm, and another with sides 4.5 cm and 3 cm. Check whether they are similar:
Ratio of lengths = 3 / 4.5 = 2/3
Ratio of breadths = 2 / 3 = 2/3
Both ratios are equal (2/3), and corresponding angles are both 90°. Therefore the two rectangles are similar.
Worked Example: Similar Triangles
Consider two right triangles with sides (3, 4, 5) and (6, 8, 10). Check their side ratios:
3/6 = 1/2 4/8 = 1/2 5/10 = 1/2
All three ratios equal 1/2, confirming the triangles are similar with a scale factor of 1:2.
Special Cases: Figures That Are Always Similar
| Figure | Always Similar? | Reason |
|---|---|---|
| All Squares | ✅ Yes | All angles = 90°; sides always in ratio 1:1 (scaled) |
| All Equilateral Triangles | ✅ Yes | All angles = 60°; all sides always proportional |
| All Circles | ✅ Yes | Shape is always the same; radii scale proportionally |
| All Rectangles | ❌ Not always | Angles equal but sides may not be proportional |
| All Triangles | ❌ Not always | Angles and side ratios may differ |
Dilation — Enlarging and Reducing Figures
Dilation produces figures that are similar (not congruent, unless the scale factor is 1). This is exactly what happens when you zoom in or out on a map, resize a photo, or build a scale model of a building.
Scale Factor Rules
| Scale Factor (k) | Effect on Figure | Example |
|---|---|---|
| k > 1 | Figure is enlarged (bigger than original) | k = 2 → each side doubles |
| 0 < k < 1 | Figure is reduced (smaller than original) | k = 0.5 → each side halves |
| k = 1 | Figure is identical to original (congruent) | No change in size |
Scale factor (k) = Side of new figure / Corresponding side of original figure
Notice that dilation always produces a similar figure because the shape is preserved — only the size changes. When k = 1, the dilation produces a congruent figure. This connects the two big ideas of this chapter: congruence is a special case of similarity where the scale factor equals 1.
Congruence vs Similarity — Key Differences
| Property | Congruent Figures (≅) | Similar Figures (~) |
|---|---|---|
| Shape | Same | Same |
| Size | Same | May differ |
| Corresponding angles | Equal | Equal |
| Corresponding sides | Equal | Proportional (ratio = scale factor) |
| Scale factor | Always 1 | Any positive value |
| Symbol | ≅ | ~ |
| Example | Two 5 cm circles | A 3 cm and a 6 cm circle |
Common Mistakes to Avoid
- Confusing similar with congruent: Similar figures look the same but may be different sizes. Only congruent figures are both the same shape and size.
- Ignoring vertex order in congruence notation: △ABC ≅ △PQR means A↔P, B↔Q, C↔R. Changing the order (e.g. △ABC ≅ △QPR) is incorrect unless the correspondence is verified.
- Assuming all rectangles are similar: Rectangles always have equal angles (90°) but their side ratios may differ — so not all rectangles are similar.
- Forgetting that reflections preserve congruence: A flipped figure is still congruent to the original — students sometimes think the mirror image is a "different" figure.
- Mixing up scale factor direction: If original → enlarged, scale factor > 1. If original → reduced, scale factor < 1. Always check which direction the dilation goes.
What This Lesson Prepares You For
The concepts you learn in this introduction — congruence, similarity, transformations, and dilation — form the foundation for every geometry topic in Class 9 and Class 10. In particular:
- Class 9 – Triangles: The formal criteria for congruence (SSS, SAS, ASA, RHS) and similarity (AA, SSS, SAS) of triangles build directly on what you learn here. See Introduction to Triangles.
- Class 10 – Similar Triangles: Theorems like Basic Proportionality Theorem (Thales) and the Pythagoras theorem use similarity extensively. See Class 10 Triangles – Introduction.
- Class 8 – Construction of Quadrilaterals: Congruence is the reason a unique quadrilateral can be constructed from five given measurements. See Construction of Quadrilaterals – Introduction.
- Class 10 – Coordinate Geometry: Reflections and rotations are described algebraically using coordinate rules, which extend the transformation ideas from this chapter.
Mastering this introduction gives you a visual language for geometry that will make every subsequent chapter easier to understand and remember.