Similarity

Introduction

• Similarity of shapes means that the shapes are same but they have different size, with equal corresponding angles and proportional sides.
• Concept of Similarity is applied in many places like linear algebra and data analysis to measure how much the two objects or datasets are similar.
• Concept of Similarity – In Geometry – To determine properties of shapes and solve problems that involve proportions.
• In Computer Science – To detect patterns and compare objects.

Similar Shapes

• Similar Shapes are those which are of same shape but have different size. The key characteristics of Similar Shapes are –
• Same corresponding Angles
• Same Shape Different size (proportional sides)

SCALE FACTOR:

• The Scale Factor tells us how much bigger the larger shape is than the smaller one.

• As the scale factor from smaller to larger shape is positive (greater than 1) we prefer using it to solve questions rather than from larger shape to smaller (lesser than 1 and create complexity).
• The following rectangles are similar because they are of same shape and the equal corresponding angles with proportional sides.

• It is same for the other sides also-

• The ratio of the corresponding sides is also same in both the shapes –

Similar Triangles

• Similar Triangles are those which are of the same shape but different size along with equal corresponding angles and proportional sides.
• The Length scale factor formula is same in case of triangles as that of the other shapes.

Solved Examples

Solved Example

If the following pair of shapes are similar then find out the missing Sides/Angles?

SOLUTION

Scale Factor Calculation: The corresponding sides given are 8cm on the larger shape and 4cm on the smaller shape.

$$\text{Scale Factor} = \frac{8}{4} = 2$$

Finding X: We know that the ratio of the corresponding side on the larger shape (4cm) to the side on the smaller shape ($X$) equals the scale factor.

$$\frac{4}{X} = 2$$

Rearranging algebraically:

$$2X = 4 \implies X = 2\text{ cm}$$

Final Answer: $X = 2\text{ cm}$

Solved Example

Find the value of Angle Y and side X?

SOLUTION

Finding Angle Y: In similar shapes, corresponding angles are equal. Therefore,

$$\angle Y = 70^{\circ}$$

Scale Factor Calculation: The correctly corresponding sides between the two shapes are 7.5cm (larger shape) and 5cm (smaller shape).

$$\text{Scale Factor} = \frac{7.5}{5} = 1.5$$

Finding X: We apply this correct scale factor to the other corresponding side pair ($X$ and 2cm).

$$\frac{X}{2} = 1.5 \implies X = 2 \times 1.5 = 3\text{ cm}$$

Final Answer: $X = 3\text{ cm}$ and $\angle Y = 70^\circ$

Solved Example

Show that the polygon ABCDE is similar to PQRST?

SOLUTION

Finding Missing Angles: The sum of interior angles in a pentagon is $540^\circ$.

$$90^{\circ} + 90^{\circ} + 90^{\circ} + 2X = 540^{\circ}$$

$$270^{\circ} + 2X = 540^{\circ} \implies 2X = 270^{\circ} \implies X = 135^{\circ}$$

Conclusion: The corresponding angles between ABCDE and PQRST are equal. However, to strictly prove similarity for pentagons, we must also establish that all corresponding side lengths are proportional. We can only state that they could be similar based on angles, but more side-length data is required for a definitive proof.

Final Answer: $X = 135^\circ$

Solved Example

Are the following triangles similar if yes find length scale factor?

SOLUTION

Yes, because the shape is same and sides are in proportion –

$$\text{Scale Factor} = \frac{4}{2} = 2$$

Final Answer: $2$

Solved Example

Are the following triangles similar? Give reason.

SOLUTION

Yes, the triangles are similar because they share equal corresponding angles (45° is explicitly given, and 90° is shown visually), and their corresponding sides are perfectly in proportion:

$$\text{Ratio} = \frac{6}{3} = \frac{2}{1} = 2$$

Final Answer: Yes, they are similar with a scale factor of 2.

Solved Example

Find out the missing sides of the triangle shown in diagram –

SOLUTION

The Triangles shown in above diagram are similar, because – According to the interior angle property of parallel lines –

$$a = e$$

$$b = f$$

Vertically Opposite Angles –

$$c = d$$

Determine Scale Factor: By tracing the alternate interior angles, the side length of 12cm on the larger bottom triangle corresponds to 6cm on the smaller top triangle.

$$\text{Scale Factor} = \frac{12}{6} = 2$$

Finding X: Side $X$ corresponds to 8cm.

$$\frac{X}{8} = 2 \implies X = 8 \times 2 = 16\text{ cm}$$

Finding Y: Side $Y$ corresponds to 7cm.

$$\frac{Y}{7} = 2 \implies Y = 7 \times 2 = 14\text{ cm}$$

Final Answer: $X = 16\text{ cm}$ and $Y = 14\text{ cm}$