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The Pythagoras theorem

The Pythagoras theorem is a fundamental theorem in Geometry. This theorem states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.

Hypotenuse of a triangle

The Hypotenuse of a right triangle is the side opposite the right angle and it is the longest side of the triangle.

In △ABC, Side AC is the hypotenuse. It is the side opposite the right angle ∟ABC.  • Pythagoras was a Greek mathematician and the word “hypotenuse” comes from the Greek word “hypoteinousa” which means "stretching under."

Pythagorean theorem calculation

The Pythagoras theorem can be written as an equation. In a right-angled triangle, if ‘a’ is the length of one side, ‘b’ is the length of the second side and ‘c’ is the length of the hypotenuse then the Pythagoras theorem states that

c2 = a2 + b2 • In a triangle, if c2 < (a2 + b2) then the triangle is an acute-angled triangle​
• In a triangle, if c2 > (a2 + b2 ) then the triangle is an obtuse angled triangle

Finding the hypotenuse

The Pythagoras theorem states that for a right angled triangle c2 = a2 + b2​. Hence, we can find the hypotenuse c as:

c = Square root of (a2 + b2)

Where a and b are the sides of the right-angled triangle.

Practice Questions

Q1: A ABC has a side of length 5cms, another side of length 12 cms and the length of its hypotenuse is equal to 13 cms. Determine if the triangle is a right angled triangle.

According to the Pythagoras theorem, in a right-angled triangle, the sum of the squares of the sides is equal to the square of its hypotenuse.

Sum of the square of the sides = 52 + 122 = 25 + 144 = 169

Square of the hypotenuse = 132 = 169

Since the two are equal, ABC is a right angled triangle.

Q2: Find the length of a hypotenuse of a right-angled triangle where the length of the other two sides is 3 cms and 4 cms.

Answer: To find the length of the hypotenuse c we can use the Pythagoras theorem

c = Square root of (a2 + b2)

= Square root of (32 + 42) = Square root of (9+16) = Square root of 25 = 5 cms

Q3: Find the length of a hypotenuse of a right angled triangle where the length of the other two sides is 6 cms and 8 cms.

To find the length of the hypotenuse we can use the pythagoras theorem

c = Square root of (a2 + b2)

= Square root of (62 + 82) = Square root of (36 + 64) = Square root of 100= 10 cms

Q4: Find the length of the third side of a right angled triangle whose one side is equal to 5 cms and the the length of the hypotenuse is equal to 13 cms.

To find the length of the hypotenuse we can use the pythagoras theorem

c2 = a2 + b2

Therefore,

132 = 52 + b2

b2 = 132 - 52

b = Square root of (169 - 25) = Square root of144 = 12 cms

Q5: Determine if PQR is a right angled triangle if its sides are 4 cms, 5 cms and 7 cms.

According to the pythagoras theorem

c2 = a2 + b2

Where c is the hypotenuse or the longest side of the right angled triangle.

Therefore,

c2 = 72 = 49

a2 + b2= 42 + 52 = 16 + 25 = 41

Since c2 <> a2 + b2 we can conclude that PQR is not a right angled triangle.

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