Class 10 Mathematics

We say that two human beings are similar when they have similar body structure and features. Similarly, two geometrical objects are called similar if they both have the same shape, or one has the same shape as the mirror image of the other. For example, all circles are similar to each other, all squares are similar to each other, and all equilateral triangles are similar to each other. On the other hand, ellipses are not all similar to each other, rectangles are not all similar to each other, and isosceles triangles are not all similar to each other.

So how do we conclude if two triangles are similar? Are there axioms that we can use to deduce similarity.

Similar triangles

• Objects that are of same shape but differ in size are said to be similar.
• Two triangles are said to be similar, if
  • their corresponding angles are equal and
  • their corresponding sides are in the same ratio (or proportion)
• If corresponding angles of two triangles are equal then they are known as equiangular triangles.

Basic Proportionality Theorem or Thales Theorem

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

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2. Criteria for Similarity of Triangles

Similarity is much like congruence, except in order for triangles to be similar, they only need to have the same shape. Formally speaking, two triangles are similar when their corresponding angles are equal and their corresponding sides are proportional.

There are three easy ways to prove similarity. These techniques are much like those employed to prove congruence–they are methods to show that all corresponding angles are congruent and all corresponding sides are proportional without actually needing to know the measure of all six parts of each triangle.

Similarity criteria for triangles

Theorem 1: If the corresponding angles of two triangles are equal, then their corresponding sides are proportional and the triangles are similar. This is known as AAA similarity criterion.

Theorem 2: If the sides of one triangle are proportional to the corresponding sides of the other triangle, then their corresponding angles are equal and the triangles are similar. This is known as SSS similarity criteria.

Theorem 3: If one angle of a triangle is equal to the corresponding angle of the other triangle and the sides including these angles are proportional, then the triangles are similar. This is known as SAS similarity criteria.

Theorem 4: When the corresponding angles of two triangles are equal, then the two triangles are similar. This is referred to as the AA similarity criterion. Strictly speaking, AA is as good as AAA, because of angle sum property of a triangle

3. Areas of Similar Triangles

Recall that two figures are similar if they both have the same shape, even if one has been enlarged or shrunk. All pairs of corresponding sides in similar figures are in the same ratio, called the scale factor.

Since this scale factor affects the length of sides, it is also called the length scale factor or linear scale factor.

But what happens to the area of a figure when we enlarge it by a length scale factor? Does it also enlarge? Is it by the same or some other related scale factor?

Theorem on areas of similar triangles

Theorem: If two triangles are similar, then the ratio of their areas is equal to the ratio of the squares of their corresponding

• sides
• altitudes
• angle bisectors
• medians

4. Pythagoras Theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras's theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle. the theorem is named after the ancient Greek mathematician Pythagoras (c. 570 – c. 495 BC). Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases. The theorem has been given numerous proofs – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.

Pythagoras’ theorem

The Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

According to this theorem, when squares are drawn on the three sides of a right angled triangle, the area of the square on the longest side is equal to the sum of the areas of the squares on the other two sides.

In other words, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

i.e., if a, b, c are the lengths of the sides of a triangle with c being the lengths of the longest side,

then, a² + b² = c²

This set of three lengths a, b, c is called a Pythagorean triplet.

Some examples of Pythagorean triplets are (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25)

Converse of Pythagoras’ theorem

In a triangle, if the square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

In other words, if the set of side lengths of the triangle form a Pythagorean triplet, then the angle opposite the longest side is a right angle.