Pythagorean Theorem Formula, Proof and Examples

Relationship between sides in a Right Triangle

The Pythagorean theorem provides an understanding of the relationship between the sides of a right triangle. Using the Pythagorean triples, students are able to solve real-world problems involving distances, measurements, and angles. The Pythagorean Theorem formula serves as a stepping stone for advanced mathematical concepts and applications, including trigonometry, calculus. and linear algebra.

Pythagorean Theorem FREE Worksheet

What are the prerequisites to learn the Pythagorean theorem?

Angle basics, Triangles, Exponents, Square roots, Basic Algebra, and Solving algebraic equations.

What is the Pythagorean Theorem?

Pythagorean Theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides(called leg)”. The sides of this triangle have been named Perpendicular, Base, and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.

Pythagorean Theorem Proof:

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

To Prove: AC² = AB² + BC²

Construction: Draw a perpendicular BD meeting AC at D.

Proof:

In △ABC and △ADB

∠A=∠A [Common angle]

∠ADB=∠ABC [Each 90 ∘ ]

△ADB∼△ABC [A−A Criteria]

Now, AB² = AD×AC ..........(1)

Similarly, In △ABC and △BDC

BC² =CD×AC ..........(2)

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

AB² +BC² =AD×AC+CD×AC

=AC(AD+CD)

=AC×AC = AC²

Hence, the Pythagorean Theorem is proved.

Can we prove the Pythagorean Theorem in Geometry?

Consider a right-angled triangle having sides a, b, and c, where a and b are the legs of a triangle and c is the hypotenuse.

Take 4 such similar triangles having sides a, b, and c and arrange them as shown.

Area of bigger square = Area of smaller square + area of 4 similar triangles

Hence, the Pythagorean Theorem is proved.

What is the Pythagorean Theorem Formula?

(Hypotenuse)² = (perpendicular)² + (Base)²

Does the converse of the Pythagorean Theorem hold true?

Yes, it states that if the length of the sides of a triangle is a, b, and c,

and a² + b² = c², then the triangle is a right-angled triangle.

What are the Applications of the Pythagorean Theorem?

The Pythagorean Theorem has its applications in various fields such as engineering, construction, navigation, and sports.

1. Architecture and Construction:

Given two straight line, the hypothesis assists with tracking down the length of the corner to corner. This application is often utilized in design, carpentry, or other actual development projects.

2. Laying out Square Angles:

The hypothesis is utilized in development to ensure structures are square. While establishing out groundwork, or developing a square corner between two walls, development laborers will set out a triangle from three ropes that compare with these lengths. In the event that the string lengths were estimated accurately, the corner inverse the triangle's hypotenuse will be a right point, so the constructors will realize they are building their walls or establishments on the right lines.

3. Navigation:

Hypothesis is helpful for two-layered route to figure out the briefest conceivable distance between two paths and can explore to that specific point. Similar standards can be utilized for air route. For example, a plane can utilize its level over the ground and its separation from the objective air terminal to track down the perfect locations to start a plunge to that air terminal.

4. Surveying:

Hypothesis is utilized to work out the steepness of slants of slopes or mountains. An assessor utilizes a telescope to glance through a proper distance toward a gauge, shaping a right point between the view of the telescope and the gauge. The assessor may then utilize the hypothesis to track down the length of the slant that covers that distance and, from that length, decide how steep it is on the grounds that he knows about both the level of the gauge and its even separation from the telescope.

5. Face Recognition in Security Camera:

The distance between the surveillance camera and the area of the individual is noted and very much projected from the perspective utilizing the idea.

6. Electrical Engineering:

Why is it important for students to learn Pythagorean Theorem?

If two sides of a right-angled triangle are given, one can find the third side.
It is useful for determining whether the given triangle is a right triangle or not.
One can determine the length of a diagonal in a rectangle and the sides of many other geometrical shapes that have right angles.
It can be used to calculate the distance between two points on a coordinate plane.
It is a building block for more complex geometric problems that students will solve in higher classes.
It is the basis of geometry, trigonometry, calculus, and other advanced math subjects.
Questions based on Pythagorean Theorem is being asked in many standardized tests such as SAT, ACT, and GRE.

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References:

https://www.tjoe.org/pub/ei8ogu9d/release/3

https://commons.wikimedia.org/wiki/File:Pythagoras-proof-anim.svg

Applications of Pythagorean Theorem in building bridges

Applications of Pythagorean Theorem in floor setting

Applications of Pythagorean Theorem in an electric motor/generator

Pythagorean Theorem FREE Worksheet