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Apollonius's theorem is a triangular theorem that is related to the length of a median of a triangle with the length of its sides. This theorem states that the two equal sides of a triangle are equal to twice or half of the sides of a triangle. The theorem is mostly used in pythagoras concept.
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Key takeaways: Apollonius's theorem, median, triangle, pythagoras theorem, Laws of cosine
Also read: Isosceles Triangle Theorems
What is Apollonius' Theorem
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The median of any triangle can be defined as the line segment that connects the vertex of a triangle with the midpoint of its opposite sides. In Apollonius Theorem, the length of the median is connected with the length of the bisected side of a triangle and its other two sides.
In other words, this theorem relates the length of the median of a triangle with its length of sides. This theorem states that the sum of the squares of two sides of any triangle is equal to the twice of its square on the half of its third side, along with the twice of its square on the median which can bisects its third side.
Apollonius Theorem Formula
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Let x, y and z be the lengths of the sides of any triangle and say d be the length of the median which bisects the triangle at x.
Let m = x/2, be its length that bisects its half.
Then the Apollonius Theorem can be stated as
y2 + z2 = 2(m2 + d2)
This is called the Apollonius Theorem Formula.
Also read: Calculus Formula
Derivation of Apollonius Theorem
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Apollonius Theorem of a triangle can be proved with the help of Law of Cosines of a triangle.
Let ABC be a triangle with the sides AB, BC, AC be a, b and c and let AE be the median cut the triangle into two sides.
Let median d make an angle say θ on the side facing a and angle θ′ on one side that faces the side c.
Then for the triangular sides of a, d and m, based on the law of Cosines,
a2 = d2 + m2 - 2md cos θ ………..(i)
And for the sides of a triangle, c, d and m we can write it as
c2 = d2 + m2 - 2md cos θ′.
But now, θ + θ′ = π = 180°.
So cos θ′ = - cos θ, that can be used in the equation connecting c, d and m.
Thus we can get the equation,
c2 = d2 + m2 + 2md cos θ ………..(ii)
Now adding the eq (i) with the eq (ii) we can get
c2 + a2 = 2 (d2 + m2)
Hence proved the Apollonius Theorem of a triangle.
Also read: Determinant Formula
Proof of Apollonius Theorem by Pythagoras Theorem
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Let ABC be a triangle and M be the midpoint of the triangle of its side BC,
To prove: AB2 + AC2 = 2 {AM2 + (BC/2)2 }.
Proof: Let AP be the perpendicular line from the vertex A to the side of the triangle BC.
So it can be written as
BC = CM = BC/2
BP + CP = BC
Now according to the pythagorean theorem,
AB2 = AP2 + BP2
AC2 = AP2 + CP2
AM2 = AP2 + MP2
Now from the above given equation we can conclude that
AB2 + AC2 = 2AP2 + BP2 + CP2
= 2AP2 + 2MP2 + BP2 - MP2 + CP2 - MP2
= 2AM2 + (BP + MP) (BP - MP) + (CP + MP) (CP - MP)
= 2AM2 + (BP + MP) BM + CM (CP - MP)
= 2AM2 + BC2/2
= 2 {AM2 + (BC/2)2 }.
Things to Remember
- The Apollonius Theorem is an elementary theorem which is somewhat similar to the Pythagoras theorem.
- The Apollonius Theorem is mainly useful in the calculation of the length of the medians of a triangle.
- This theorem is also similar to the Parallelogram law. This can also be used in finding the length of one diagonal parallelogram if the other diagonal and its other two sides are known.
- It is also a special case of the Stewart’s Theorem that mainly deals with the more general situation of the triangle like cevian.
- A cevian, similar to a median, is a type of line segment which connects one vertex and the opposite sides of a triangle.
Also read: Differentiation and Integration Formula
Sample Questions
Ques. Suppose a triangle ABC has sides 7, 6 and 10 cm then calculate the length of the median to the side of length 10 cm? (3 marks)
Ans. It is given in the question that the triangle ABC has three sides a, b, and c.
Then Let a=10 cm,b=7 cm,c=6 cm.
As we know that the side a = 10 cm is bisected into two equal parts, we have, m = a/2 = 5 cm.
Suppose the length of the median be given by d.
According to Apollonius Theorem Formula,
c2 + b2 = 2 (m2 + d2)
Now substituting the required values, we get, 62 + 72 = 2 (52 + d2)
36 + 49 = 2 (25 + d2)
85 – 50 = 2 d2
2d2 = 35
d2 = 35/2
d = √35/2
d = 4.183 cm
Therefore, the length of the median of the triangle is d = 4.183 cm.
Ques. What is Apollonius' Theorem? (3 marks)
Ans. The Apollonius Theorem is a theorem that connects the lengths of the sides, with the length of the median of a triangle. In other words, we can say that the sum of the squares of two sides of any triangle is equal to the twice of its square on the half of its third side, along with the twice of its square on the median which can bisects its third side.
Mathematically, let AB, AC be the side lengths of a triangle and AD be a median of the side BC. Then we can write it as
AB2 + AC2 = 2 (AD2 + BD2)
This is known as Apollonius Theorem.
Ques. How is the median of a triangle different from an angle bisector of a triangle? (2 marks)
Ans. The median of a triangle is the line segment drawn from one of the vertices of the triangle to its opposite sides, which can bisects the opposite side of a triangle. The angle bisector of a triangle is the line from one if the vertex of a triangle to its opposite sides which can bisects the angle at its vertex. However, both of them are not same unless the vertices of both are on equal sides i.e it is an isosceles triangle.
Ques. Suppose a triangle PQR has sides 3, 4 and 8 cm then calculate the length of the median to the side of length 8 cm? (3 marks)
Ans. As given in the question, the triangle PQR has sides 3 cm, 4cm and 8cm respectively.
Then let a = 8 cm, b = 4 cm, c = 3 cm
As we know that the side 8cm is bisected into two equal parts .
Then m = 8/2 = 4 cm
Now, suppose the length of the median is given by d.
According to Apollonius Theorem Formula,
c2 + b2 = 2 (m2 + d2)
Now substituting the required values, we get,
32 + 42 = 2(42+ d2)
9 + 16 = 2( 8 + d2)
17 = 2 × 8 + 2 × d
17= 16 + 2d2
17 - 16 = 2d2
2d2 = 1
d2 = ½
d = √ ½
d = ¼
d = 0.25cm
Therefore the length of the median of the triangle is 0.25 cm .
Ques. The sides of a parallelogram are 12cm and 8 cm and one of its diagonals is 16 cm long. Find the length of the other diagonal. (3 marks)
Ans. As we know about a parallelogram, if it is cut into halves along the side of its diagonal then the two halves form a triangle.
And the other diagonal is a median which bisects the first diagonal and vice versa.
Let a = 16 cm, b = 12 cm and c = 8 cm.
m = a/2 = 16/2 = 8 cm
According to the formula for Apollonius Theorem we have,
c2 + b2 = 2 (m2 + d2)
Now substituting the required values, we get,
122 + 82 = 2(82 + d2)
144 + 64 = 2(64 + d2)
208 = 128 + 2d2
2d2 = 208 - 128
2d2 = 80
d2 = 80/2
d2 = 40
d = √ 40
d = 6.32 cm
Thus the median of the triangle is 6.32cm but the diagonal is actually double the value of the median.
Therefore the other diagonal = 2 × 6.32 cm = 12.64 cm.
Ques. Give the relation between Pythagoras’ Theorem and Apollonius’ Theorem? (3 marks)
Ans. Pythagoras theorem is a theorem which is used in the case of right angled triangles and Appollonius theorem is applicable for all triangles. However in the case of isosceles triangles pythagoras theorem. Pythagoras’ Theorem can be seen as a special case of Apollonius’ Theorem, when we have an isosceles triangle.
Let AP be the perpendicular line from the vertex A to the side of the triangle BC.
So it can be written as
BC = CM = BC/2
BP + CP = BC
Now according to the pythagorean theorem,
AB2 = AP2 + BP2
AC2 = AP2 + CP2
AM2 = AP2 + MP2
Now from the above equation we can conclude that
AB2 + AC2 = 2AP2 + BP2 + CP2
= 2AP2 + 2MP2 + BP2 - MP2 + CP2 - MP2
= 2AM2 + (BP + MP) (BP - MP) + (CP + MP) (CP - MP)
= 2AM2 + (BP + MP) BM + CM (CP - MP)
= 2AM2 + BC2/2
= 2 {AM2 + (BC/2)2 }.
This is simply Pythagoras’ Theorem in the right angle triangle.
Ques. Find out the other diagonal of a parallelogram if the sides of the parallelogram are given as 10 cm and 15cm, and the diagonal is 20 cm. (3 marks)
Ans. Let a = 20 cm, b = 15 cm and c = 10 cm.
m = a/2 = 20/2 = 10 cm
According to the formula for Apollonius Theorem we have,
c2 + b2 = 2 (m2 + d2)
Now substituting the required values, we get,
102 + 152 =2(102 + d2)
100 + 225 = 2(100 + d2)
325 = 200 + 2d2
325 - 200 = 2d2
125 = 2d2
d2 = 125/2
d2 = 62.5
d = √62.5
d = 7.90 cm
Thus the median of the triangle is 7.90 cm but the diagonal is actually double the value of the median.
Therefore other diagonal = 2 × 7.90 cm= 15.80cm
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