Apollonius Theorem formula proof and how to solve problems in triangles
It is understood that theorems in mathematics refer to the statement of a result that has been proven based on previously set statements, such as theorems, or statements commonly acknowledged, such as axioms. In mathematics, theorems are defined as the results derived from a set of axioms that are found to be accurate. There are some axioms in mathematics that are considered mathematical logic, but with systems formulated as questions or statements.
Apollonius' Theorem
Named after the 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. It states that the sum of squares of any two sides of a triangle is equal to twice of its square on half of the third side, along with the twice of its square on the median that bisects the third side. 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 and Proof of Apollonius' Theorem
There is no doubt that medians make up the most important set of components in the geometry of triangles and they can be regarded as independent of the geometric shapes of the triangles. According to Apollonius' Theorem, the sides and the medians of the triangle are related. Apollonius' theorem refers to the relationship between the lengths of the sides of a triangle and the length of its median.
Apollonius’ Theorem Statement
"The sum of the squares on two sides of a triangle equals the sum of the squares on one half of the third side, plus the sum of the squares along the median of the third side"
OR
The midpoint of any triangle LMN is O, so the formula (LM)²+ (LN)²= 2 \[\left [ (LO)² + (MO)² \right ]\] = O.
Apollonius’ Theorem Proof
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Consider that O is the origin of the rectangular form and MN and OY represent the x-axis and y-axis, respectively, of the Cartesian coordinates. When MN = 2a, the coordinates of points M, as well as N, are (a, 0) and (-a, 0), respectively. When L coordinates are (b, c), then
Since the coordinates of the point O are [0, 0], LO2 = (c – 0)2 + (b – 0)2 .
= c² + b²;
In other words, LM2 = (c – 0)² + (b + a)² = c² + (a + b)²
MO² = (0 – 0)² + (- a – 0)² = a²
LN² = (c – 0) ² + (b – a) ² = c² + (a – b)²
In other words, LN2 + LM2 = c2 + (a + b)² + c² + (b – a)²
= 2c² + 2 (a² + b²)
= 2(b² + c²) + 2a²
= 2LO² + 2MO²
= 2 (LO² + MO²).
= 2(MO² + LO²). Hence Proved.
Statement and Proof by the Pythagorean Theorem
Statement: For a triangle ABC with M be the midpoint of its side BC,
AB² + AC² = 2{AM² + (BC/2)²}
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Proof: Let AH be the perpendicular from A on BC
So, it is clear that
BM = CM = BC/2
BH + CH = BC
Now, use the Pythagorean Theorem
AB² = AH² + BH²
AC² = AH² + CH²
AM² = AH² + MH².
From the above equations, we can conclude that:
AB² + AC² = 2AH² + BH² + CH²
= 2AH² + 2MH² + BH² − MH² + CH² − MH²
= 2AM² + (BH + MH) (BH−MH) + (CH+MH) (CH−MH)
= 2AM² + (BH+MH) ⋅ BM + CM ⋅ (CH−MH)
= 2AM² + BC²/2
= 2 (AM² + (BC/2)²). Hence, proved
Statement and Proof by 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∣
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AB² + AC² = ∣b∣²+ ∣c∣²
=1/2 (2∣b∣²+ 2∣c∣²)
= ½ (∣b∣² + ∣c∣² + 2b⋅c + ∣b∣² + ∣c∣²− 2b⋅c)
= ½ {(b+c)² + (c−b)²}
= ½ (4AM²+ BC2)
= 2 (AM² + (BC/2)²) Hence Proved.
FAQs on Apollonius Theorem in Triangle Geometry
1. What is Apollonius Theorem in geometry?
The Apollonius Theorem states that in any triangle, the sum of the squares of two sides equals twice the square of the median plus half the square of the third side.
For triangle ABC with median AD to side BC:
AB² + AC² = 2(AD²) + (1/2)BC²
This theorem relates the median of a triangle to its three sides and is widely used in coordinate geometry and triangle proofs.
2. What is the formula for Apollonius Theorem?
The formula for Apollonius Theorem is AB² + AC² = 2(AD²) + (1/2)BC², where AD is the median to side BC.
- AB and AC are two sides of the triangle.
- BC is the third side.
- AD is the median from vertex A.
3. How do you prove Apollonius Theorem?
Apollonius Theorem is proved using the Pythagoras Theorem on two smaller triangles formed by the median.
Steps:
- Let AD be the median of triangle ABC.
- Since D is the midpoint, BD = DC.
- Apply Pythagoras Theorem in triangles ABD and ACD.
- Add the two equations obtained.
4. How do you find the length of a median using Apollonius Theorem?
The length of a median is found using AD = √[(2AB² + 2AC² − BC²)/4].
Example:
- Let AB = 5, AC = 7, BC = 8
- AD² = (2(5²) + 2(7²) − 8²)/4
- AD² = (50 + 98 − 64)/4 = 84/4 = 21
- AD = √21
5. Is Apollonius Theorem related to the Pythagoras Theorem?
Yes, Apollonius Theorem is a generalization of the Pythagoras Theorem.
In a right triangle:
- The median to the hypotenuse equals half the hypotenuse.
- Apollonius Theorem reduces to the Pythagorean relation.
6. Can you give an example of Apollonius Theorem?
Yes, for a triangle with sides 6, 8, and 10, the median to side 10 can be found using Apollonius Theorem.
- AB = 6, AC = 8, BC = 10
- AD² = (2(6²) + 2(8²) − 10²)/4
- AD² = (72 + 128 − 100)/4 = 100/4 = 25
- AD = 5
7. What is the importance of Apollonius Theorem?
The importance of Apollonius Theorem lies in its ability to relate the sides of a triangle with its median.
- Helps find unknown median lengths.
- Used in coordinate geometry problems.
- Assists in proving other triangle properties.
- Connects to centroid and vector formulas.
8. What is the difference between Apollonius Theorem and the median formula?
The median formula is derived from Apollonius Theorem and gives a direct expression for the median length.
- Apollonius Theorem: AB² + AC² = 2AD² + (1/2)BC²
- Median Formula: AD = √[(2AB² + 2AC² − BC²)/4]
9. Does Apollonius Theorem apply to all triangles?
Yes, Apollonius Theorem applies to all types of triangles—acute, obtuse, and right-angled.
Conditions:
- The figure must be a valid triangle.
- The segment considered must be a median (joining a vertex to midpoint).
10. What are common mistakes when using Apollonius Theorem?
The most common mistake in Apollonius Theorem is confusing the median with other line segments like altitude or angle bisector.
- Using a side that is not a midpoint division.
- Forgetting the 1/2 BC² term.
- Incorrect squaring of side lengths.
- Arithmetic errors while simplifying.
