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.

### Frequently Asked Question’s

**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.