Applying Pythagoras’ Theorem – Proof, Formula | Questions on Applying Pythagorean Theorem

Pythagora’s theorem states that in a right-angled triangle the sum of the square of the two sides is equal to the square of the hypotenuse. This theorem is used to state whether the given triangle is a right triangle or not. Some of the real-time applications of the Pythagorean theorem is provided here. Get the proof of applying Pythagora’s theorem and solved problems on this page.

State & Proof Applying Pythagoras’ Theorem

Applying Pythagorean theorem states that if in ∆PQR, Q at a right angle, M and N are the midpoints of the sides PQ, QR, then PN² + RM² = 5MN².

Proof:
Let us take ∆PQR where ∠PQR = 90°.
Point M, N the midpoints of sides PQ, QR.
So, PM = MQ and QN = NR
Therefore, PQ = 2MQ and QR = 2QN
To prove: PN² + RM² = 5MN²

Statement Reason
∆PQN, PQ² + QN² = PN²
(2MQ)² + QN² = PN²
4MQ² + QN² = PN²
By Pythagorean Theorem
PQ = 2MQ
∆RQM, MQ² + QR² = RM²
MQ² + (2QN)² = RM²
MQ² + 4QN² = RM²
By Pythogoras theorem
Given
5MQ² + 5QN² = PN² + RM²
5(MQ² + QN²) = PN² + RM²
Adding statements 1 and 2
5MN² = PN² + RM² Applying Pythagoras’ theorem in ∆QMN

Hence, proved.

Applications of Pythagoras Theorem

The applications of the Pythagorean theorem are listed here.

• This Pythagoras theorem is commonly used to determine the lengths of sides of a right-angled triangle.
• The theorem is used to find the diagonal length of a rectangle, square, others.
• It is used in security cameras for face recognition.
• This theorem is applied in surveying the mountains.
• It is used in trigonometry to get the trigonometric ratios like sin, cos, tan, cos, sec, cot.
• It is also used in navigation to find the shortest route.
• By using the Pythagoras theorem, we can derive the formula, base, hypotenuse and perpendicular.
• The theorem is used to calculate the steepness of slopes of hills or mountains.

Questions on Applying Pythagoras Theorem

Question 1:
In a right angle triangle ABC, ∠ABC = 90°, X and Y are the midpoints of the base and perpendicular sides. If AX = 5 cm, BY = 8 cm, find XY.

Solution:
Given that,
AX = 5 cm, BY = 8 cm
ABC is a right angle triangle, X and Y are the midpoints of the base and perpendicular sides
According to applying Pythagoras theorem AX² + BY² = 5XY²
Substitute the given values.
5XY² = 5² + 8²
5XY² = 25 + 64 = 89
XY² = 17.8
XY = 4.219
Therefore, XY = 4.219 cm.

Question 2:
If △PQR is a right triangle, M and N are the midpoints of two legs, the lengths of RM = 15 cm, MN = 20 cm. Find the length of PN.

Solution:
Given that,
△PQR is a right triangle
M and N are the midpoints of PQ and QR
RM = 15 cm, MN = 20 cm
By using applying Pythagoras theorem
PN² + RM² = 5MN²
PN² + 15² = 5(20)²
PN²+ 225 = 5(400)
PN² + 225 = 2000
PN² = 2000 – 225 = 1775
PN = 42.13
Therefore, the length of PN is 42.13 cm.

1. What is meant by the Pythagoras theorem?

The Pythagoras theorem states that in a right triangle, the sum of the square of the base and perpendicular is the square of the hypotenuse.

2. In which type of triangle does the Pythagoras theorem hold?

Right triangles hold the Pythagoras theorem.

3. What is the significance of the Pythagorean theorem?

After discovering Pythagorean theorem, Greeks prove that the existence of numbers that can’t be expressed as rational numbers. In a right triangle with a base, perpendicular 1 unit each is constructed then hypotenuse is √2 which is not a rational number.

4. Where did Pythagoras theorem come from?

The Pythagoras theorem was first discovered by the mathematician named Pythagoras.