How do you prove Apollonius theorem?

Apollonius’ theorem, in general, is proved to be correct by using coordinate geometry, but it can also be proved by using the Pythagorean theorem and vectors. Now, let’s go through the statement and proof of this theorem. Statement: If O is the mid-point of a side MN of any triangle LMN, then LM² + LN² = 2(LO² + MO²).

What is corollary in the Pythagoras Theorem?

The Pythagorean equation relates the sides of a right triangle in a simple way, so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides, but less than their sum.

What is Pythagoras theorem Class 8?

Pythagoras’ theorem states that for all right-angled triangles, ‘The square of the hypotenuse is equal to the sum of the squares of the other two sides’. The hypotenuse is the longest side and it’s always opposite the right angle.

What is meant by Apollonius theorem?

In geometry, Apollonius’s theorem is a theorem relating the length of a median of a triangle to the lengths of its sides. It states that “the sum of the squares of any two sides of any triangle equals twice the square on half the third side, together with twice the square on the median bisecting the third side”.

What is use of Apollonius theorem?

Apollonius’ Theorem is a theorem in elementary geometry, similar to Pythagoras Theorem. It is useful to calculate the lengths of a median of a triangle. It is equivalent to the Parallelogram Law, as stated before.

What is Pythagorean theorem Class 10?

Pythagoras 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”. The sides of the right-angled triangle are called base, perpendicular and hypotenuse .

What is the Apollonius theorem in math?

A. Apollonius theorem also known as Median of a triangle theorem states that in triangle ABC, if AB, BC and AC are the sides and AD is the median to BC, then AB2 + AC2 = 2 (AD2 + BD2).

How to prove that triangle ABC satisfies Apollonius’s theorem by using vectors?

Statement: For a triangle, ABC having M as the midpoint of side BC, AB2 + AC2 = 2 (AM2 + (BC/2)2), i.e., triangle ABC satisfies Apollonius’s theorem by using vectors. Proof: Let A be the Cartesian coordinate of triangle ABC and define AB = ∣b∣ and AC = ∣c∣, then it is clear that AM = (b+c)/2​ and BC = ∣c∣ – ∣b∣ = 2 (AM2 + (BC/2)2) Hence Proved.

What is the purpose of the circle of Apollonius?

The circle of Apollonius is named after the ancient geometrician Apollonius of Perga. This beautiful geometric construct can be helpful when solving some general problems of geometry and mathematical physics, optics, and electricity. Here we discuss two of its applications: localizing an object in space and calculating electric fields.

What is the step-by-step process for proving a theorem called?

The step-by-step process for proving a theorem to be accurate is called proof. Named after a Greek mathematician Apollonius, this theorem is an elementary theorem that relates the length of a median of any triangle to the lengths of its edges.