Question

Define Pythagoras Theorem, its history, detailed proof, converse, Uses of the theorem, application and its consequences. Draw diagrams or paste pictures wherever required.

Answer 1

What is the formula for Pythagorean Theorem?

The formula for Pythagoras, for a right-angled triangle, is given by; c2=a2+b2

What is the formula for hypotenuse?

The hypotenuse is the longest side of the right-angled triangle, opposite to right angle, which is adjacent to base and perpendicular. Let base, perpendicular and hypotenuse be a, b and c respectively. Then the hypotenuse formula, from the Pythagoras statement will be;

c = √(a2 + b2)

Can we apply the Pythagoras Theorem for any triangle?

No, this theorem is applicable only for the right-angled triangle.

What is an example of Pythagoras theorem?

An example of using this theorem is to find the length of the hypotenuse given the length of the base and perpendicular of a right triangle.

What is the use of Pythagoras theorem?

The theorem can be used to find the steepness of the hills or mountains. To find the distance between the observer and a point on the ground from the tower or a building above which the observer is viewing the point. It is mostly used in the field of construction.

Answer 2

Tip:

• Pythagoras Theorem can be proved in several ways.

Explanation:

• We have to define Pythagoras theorem, its history, detailed proof, converse, Uses of the theorem, application and its consequences.

Step

Step 1 of 7:

Pythagoras Theorem:

The Pythagoras theorem states that if a triangle is right-angled, then the square of the hypotenuse is equal to the sum of squares of other two sides.

Step 2 of 7:

History of Pythagoras theorem:

The Pythagorean theorem was first known in ancient Babylon and Egypt (beginning about 1900 B.C.). The relationship was shown on a 4000 year old Babylonian tablet now known as Plimpton 322. However, the relationship was not widely publicized until Pythagoras stated it explicitly.

Step 3 of 7:

Proof of Pythagoras theorem:

Given: A right-angled triangle $ABC$, right-angled at $B$.

To Prove: $AC^2=AB^2+BC^2$

Construction: Draw a perpendicular $BD$ meeting $AC$ at $D$.

Proof:

We know, Δ$ADB$ $\sim$Δ$ABC$

Therefore,  $\frac{AD}{AB}=\frac{AB}{AC}$  (corresponding sides of similar triangles)

Or,$AB^2=AD\times AC~~~~~~~~~~~~~~~~~~~...(1)$  

Also, Δ$BDC$ $\sim$Δ$ABC$

Therefore, $\frac{CD}{BC}=\frac{BC}{AC}$   (corresponding sides of similar triangles)

Or, $BC^2=CD\times AC~~~~~~~~~~~~~~~~~~~~~...(2)$

Adding the equations (1) and (2) we get,

$AB^2+BC^2=AD\times AC+CD\times AC$

$AB^2+BC^2=AC(AD+CD)$

Since, $AD+CD=AC$

Therefore, $AC^2=AB^2+BC^2$

Hence, the Pythagorean theorem is proved.

Step 4 of 7:

Converse of Pythagoras Theorem:

The converse of Pythagoras theorem states that if the square of the third side of a triangle is equivalent to to the sum of its two shorter sides, then it must be a right triangle.

Step 5 of 7:

Use of Pythagoras theorem:

Pythagoras theorem is useful to find the sides of a right-angled triangle. If we know the two sides of a right triangle, then we can find the third side.

Step 6 of 7:

Application of Pythagoras Theorem:

• To know if the triangle is a right-angled triangle or not.

• In a right-angled triangle, we can calculate the length of any side if the other two sides are given.

• To find the diagonal of a square.
• It is useful technique for construction.
• In oceanography, the formula is used to calculate the speed of sound waves in water.
• In two-dimensional navigation, the theorem is used to calculate the shortest distance between given points.

Step 7 of 7:

Consequences:

1.   Pythagorean Triplet: A Pythagorean triple consists of three positive integers a, b, and c, such that . In other words, a Pythagorean triple represents the lengths of the sides of a right triangle where all three sides have integer lengths. Such a triplet is commonly written (a, b, c), and a well-known example is (3, 4, 5).

2.  Irrational Numbers: One of the consequences of the Pythagorean theorem is that irrational numbers, such as the square root of two, can be constructed. A right triangle with legs both equal to one unit has hypotenuse length square root of two.