Right Triangles

First Glance

A right-angled triangle is a triangle in which one angle measures 90°. It is a geometric figure with three sides, where two of the sides meet at a right angle. Right triangles are important in many areas of geometry and have been in application for several years.

The Pythagorean Theorem or Pythagoras theorem is named after the Greek mathematician Pythagoras who developed a formula to find the lengths of the sides of any right triangle. He treated each side of a right triangle as though it were a square and concluded that the total area of the two smaller squares is equal to the area of the largest square.  The theorem can be written as an equation:

a² + b² = c²

Where the largest side of a right angled triangle is represented by symbol “c” and  is called Hypotenuse and the other two smaller sides are pointed out by symbol “a” and “b” which are  referred as legs of  a triangle.  The side opposite the right angle is always the Hypotenuse. The Pythagorean Theorem studies the relation of the three sides of a right-angled triangle.

More Detail

In your earlier classes, you have come across a right triangle. It’s important to understand about the properties of a right triangle owing to its problem-solving attributes and applications in our practical life.

 

A right-angled triangle is a triangle in which one angle is equal to 90°. A right triangle can also be an isosceles triangle if its two other sides (the arms) are equal. In a right isosceles triangle, one angle is 90° and the other two angles measure 45° each.  Models of these type of triangles are available in the market and is widely used by architects, engineers, graphic artists, and carpenters in their design and construction work.

There are other forms of right triangle where one angle is equal to 90 degree and the other two angles measure 30° and 60° respectively. Here the Hypotenuse is twice as long as the shortest side. This version of triangle is also manufactured in plastic models and is used in design, drawing, and building applications. In a right-angled triangle the two angles other than the right angle can be of any measure but should always add up to 90°.

Pythagoras a Greek mathematician, who lived about 2500 years ago, developed the most famous Pythagorean Theorem, although it is often argued that knowledge of the theory predates him. He proved that, in any right triangle, the square of the Hypotenuse is equal to the sum of the side opposite sides.

The Pythagorean or Pythagoras Theorem can be written in an equation:

a² + b² = c² where c stands for the length of the hypotenuse, and a and b stand for the lengths of the other two sides.

By now, we have learnt the Pythagorean Theorem. Let’s work the Pythagorean Theorem using this right triangle with sides of 3 and 4 cm, and a hypotenuse of 5 cm. By substituting the value in the given equation a² + b² = c² we can verify this theorem.

Thus, by the Pythagoras Theorem:
a² + b² = c², where
or, 42 + 32 = 74
or, 16 + 9 = 25
or, c² = 25
or, c = √25
or, c = 5

By the help of Pythagorean Theorem you can prove whether a given triangle is right triangle or not. If one side of a right triangle is missing, you can find it using the other two sides.  Let’s try it out:

n the above right triangle ABC where AB = 5 and BC = 12 the value of AC is missing.  The side BC stands for the hypotenuse and AB and AC are the other two sides of a triangle. Using the Pythagoras Theorem a² + b² = c², we can find the length of AC<

Thus, AC² + 52 = 122
or,  AC² + 25 = 144
or, AC² = 144 – 25
or, AC = √119
or, AC = 10.9

Examples

Using the Pythagoras Theorem, find the Hypotenuse (which is denoted by “c”) in the above triangle.

According to Pythagoras Theorem, we know:

10² + 6² = AC²
or, 100 + 36 = AC²
or, 136 = AC²
or, √136 = AC
or, 11.67 = AC

Thus, the length of the Hypotenuse AC in the triangle ABC = 11.67

Using the Pythagoras Theorem, verify whether the following sides would make up a right-angled triangle. The sides of the triangle are a = 4, b = 5, c = 6.4

As, a² + b² = c²
or, 42 + 52 = c²
or, 16 + 25 = c²
or, 41 = c²
or, √41 = c
or, 6.4 = c

Thus, we can say that the given sides would make up a right-angles triangle.

Test your knowledge – try our Right Triangles Test.