Similar figures are shapes that have the same shape but not necessarily the same size.
- All corresponding angles are equal.
- The ratios of corresponding sides are equal (one shape is a scaled version of the other).
- It is represented using the symbol ∼.
If two triangles ABC and LMN are similar, we write it as ∆ABC ∼ ∆LMN. In similar triangles, corresponding sides are in the same ratio, and corresponding angles are equal.
Properties of Similar Figures
In mathematics, properties help us identify similar figures and understand how they relate to each other.
Corresponding Angles
Corresponding angles are the angles formed when a transversal cuts two lines, and the angles lie in the same relative position at each intersection.
If the two lines are parallel, then the corresponding angles are equal. If the lines are not parallel, the corresponding angles are not equal.
According to the diagram above, Line 1 and Line 2 are parallel, and Line 3 is a transversal. When a transversal intersects two parallel lines, the angles formed in the same relative position are called corresponding angles.
Here, ∠1 and ∠2 occupy the same position, so they are corresponding angles and are equal
Corresponding Sides
Corresponding sides are the sides that lie in the same position in two figures. In congruent figures, corresponding sides are equal, and in similar figures, corresponding sides are in the same ratio.
Let us consider 2 quadrilaterals, ABCD and PQRS, to understand the corresponding sides.
From the above diagram, we can observe that:
- The side AB corresponds to the side PQ
- The side BC corresponds to the side QR
- The side CD corresponds to the side RS
- The side DA corresponds to the side SP
Similarity for Triangle
If two triangles have their corresponding sides in the same ratio and their corresponding angles equal, then the triangles are similar. Similar triangles may differ in size, but they have equal angles and proportional sides. These properties help us understand the relationship between sides and angles using triangle similarity theorems.
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Solved Examples
Example 1: In the Δ ABC length of the sides is given as AP = 4 cm, PB = 12 cm, and BC = 20 cm. Also PQ||BC. Find PQ.
Solution:
Given: AP = 4cm, PB = 12cm, BC = 20cm
AB = 4+12 = 16cm
Since PQ||BC ⇒ ΔAPQ~ΔABC
AP/AB = PQ/BC
4/16 = PQ/20
1/4 = PQ/20
PQ = 20/4 = 5cm
Example 2: If the following figures are similar, Find the value of x.
Solution: 5/20 = 12/x = 13/52
5×x = 20×12
5x = 240
5x/5 = 240/5
x = 48
Practice Questions
Ques 1: In the ΔABC length of the sides is given as AP = 8 cm, PB = 20 cm, and BC = 9 cm. Also PQ||BC. Find PQ.
Ques 2: In the ΔABC length of the sides is given as AP = 12 cm, PB = 30 cm, and BC = 10 cm. Also PQ||BC. Find PQ.