How to Find the Length of a Median From the Sides of a Triangle
The relation between the median and the side of a triangle connects the length of a median to the triangle’s side lengths using a precise algebraic formula. This relation is fundamental for triangle geometry and standard in JEE Main.
Mathematical Expression for Median in Terms of Triangle Sides
Consider triangle $ABC$ with side lengths $a$, $b$, and $c$. Let $m_a$ denote the length of the median from vertex $A$ to the midpoint of side $BC$. The standard relation is:
$m_a = \sqrt{\dfrac{2b^2 + 2c^2 - a^2}{4}}$
Definition: The median of a triangle is the line segment from a vertex to the midpoint of the opposite side.
For medians $m_b$ and $m_c$ from $B$ and $C$, the analogous formulas are:
$m_b = \sqrt{\dfrac{2a^2 + 2c^2 - b^2}{4}}$, $m_c = \sqrt{\dfrac{2a^2 + 2b^2 - c^2}{4}}$
These relations are valid for all triangles. For isosceles or equilateral triangles, the expressions simplify.
Derivation Using Apollonius’s Theorem
Let $D$ be the midpoint of $BC$ in triangle $ABC$. By the distance formula and properties of medians, the length $AD$ satisfies:
$AD^2 = \dfrac{2AB^2 + 2AC^2 - BC^2}{4}$
Substitute $AB = c$, $AC = b$, $BC = a$ to obtain:
$AD^2 = \dfrac{2b^2 + 2c^2 - a^2}{4}$
Therefore, $m_a = AD = \sqrt{\dfrac{2b^2 + 2c^2 - a^2}{4}}$ as required.
For additional context on the structure of triangles and associated heights, refer to Properties Of Triangle And Height.
Algebraic Relation Among Sides and Medians of a Triangle
The medians $m_a$, $m_b$, and $m_c$ and side lengths $a$, $b$, $c$ satisfy:
$a^2 + b^2 + c^2 = \dfrac{4}{3}(m_a^2 + m_b^2 + m_c^2)$
Result: Three times the sum of the squares of the sides equals four times the sum of the squares of the medians.
Geometric Properties Involving Triangle Medians
Key geometric facts about medians include:
- Each triangle has three medians
- Medians are concurrent at the centroid
- Centroid divides each median in a ratio $2:1$
- Median bisects the side it meets
- Median divides the triangle into two equal areas
- In equilateral triangle, all medians are equal
- Medians from equal angles in isosceles triangles are equal
For area-based relations, see the Area Of Triangle Formula resource.
Calculating Medians in Specific Triangle Types
In an equilateral triangle of side $a$, the median $m = \dfrac{\sqrt{3}}{2}a.$ Thus, all medians are congruent.
For a right-angled triangle, the median to the hypotenuse equals half the hypotenuse. If $PR$ is the hypotenuse, the median $QL = \dfrac{1}{2}PR$.
This property simplifies calculations in specific exam cases. For explorations of right triangle properties, refer to Height And Distance.
Representative Problems Involving Median–Side Relations
Example: In $\triangle PQR$, let $PQ = 12$ cm, $QR = 14$ cm, and $QL$ be a median to hypotenuse $PR$. Find $QL$.
First, use Pythagoras: $PR^2 = PQ^2 + QR^2 = 144 + 196 = 340 \implies PR = \sqrt{340}$
As the median to the hypotenuse, $QL = \dfrac{1}{2}PR = \dfrac{1}{2}\sqrt{340} = \sqrt{85} \approx 9.22$ cm.
For similar practice, review Relation Between Median And Side.
Solution: Direct application of the right triangle median property yields the answer.
Example: In $\triangle ABC$, $a=13$, $b=14$, $c=15$. Find $m_a$.
Use $m_a = \sqrt{\dfrac{2b^2 + 2c^2 - a^2}{4}}$
Substitute: $= \sqrt{\dfrac{2\times14^2 + 2\times15^2 - 13^2}{4}}$
$= \sqrt{\dfrac{392 + 450 - 169}{4}} = \sqrt{\dfrac{673}{4}}= \dfrac{\sqrt{673}}{2} \approx 12.98$ units.
Exam Tip: Always use correct side assignment with respect to the chosen median.
For questions requiring area computation via medians, see Area Of Triangle Formula.
Common Errors and Identification in Median–Side Calculations
Common Error: Interchanging sides during substitution in the formula causes incorrect calculation of the median length.
Incorrectly applying properties of medians from special triangles (e.g., treating all medians of scalene triangles as equal) is a frequent source of mistakes.
The relation between medians and sides of similar triangles is not direct; median ratios follow the similarity ratio.
For further practice and comprehensive triangle concepts, refer to Area Of A Rhombus Formula.
FAQs on Understanding the Relationship Between the Median and Sides of a Triangle
1. What is the relation between the median and sides of a triangle?
The median of a triangle relates to its sides using the median length formula. For a triangle with sides a, b, and c, the length of the median m_a from vertex A (opposite side a) is:
m_a = (1/2) × sqrt(2b^2 + 2c^2 - a^2)
Key points:
- Each median connects a vertex to the midpoint of the opposite side.
- The formula helps calculate the median's length if you know the side lengths.
- The medians intersect at the centroid, dividing each median in a 2:1 ratio.
2. How do you calculate the length of the median of a triangle given the side lengths?
To find the median length from a vertex to the midpoint of the opposite side, use:
m_a = (1/2) × sqrt(2b^2 + 2c^2 - a^2)
Where:
- a is the side opposite the median
- b and c are the other two sides
- This formula is derived from the Apollonius theorem
3. What is the importance of medians in triangles?
Medians play a crucial role in triangle geometry because:
- They intersect at the centroid, which is the triangle's center of gravity
- They divide the triangle into six smaller triangles of equal area
- The centroid divides each median in a 2:1 ratio
- The median length can help determine side lengths and prove properties
4. State and explain Apollonius' theorem for triangles.
Apollonius' theorem relates medians and sides of a triangle. It states:
- For any triangle, the sum of the squares of any two sides equals twice the square of the median to the third side plus half the square of the third side.
AB^2 + AC^2 = 2AM^2 + (1/2)BC^2, where AM is the median from A to BC.
5. What is the centroid and how is it related to medians?
The centroid is the point where the three medians of a triangle meet.
- It divides each median in the ratio 2:1 (vertex to centroid is twice centroid to midpoint)
- The centroid is also the triangle’s center of mass
- It always lies inside the triangle
6. Is the median always shorter than the longest side of a triangle?
Yes, the median from any vertex is always shorter than the longest side of the triangle.
- This is because the median connects a vertex to the midpoint, not the full length of the opposite side
- Geometrically, the straight side between two triangle vertices is always the greatest distance
7. How do you find the median length in an equilateral triangle?
In an equilateral triangle with side a, all medians have equal length.
The formula is:
Median = (a × sqrt(3))/2
- This formula uses the Pythagorean Theorem
- Each median also acts as an altitude and angle bisector
8. What is the formula for all three medians of a triangle?
Given sides a, b, c, medians from each vertex are:
- m_a = (1/2) × sqrt(2b^2 + 2c^2 - a^2)
- m_b = (1/2) × sqrt(2a^2 + 2c^2 - b^2)
- m_c = (1/2) × sqrt(2a^2 + 2b^2 - c^2)
9. Can a triangle have medians of equal length even if all sides are not equal?
A triangle usually has equal medians only if it is equilateral, where all sides are equal.
- In some special isosceles triangles, two medians may be equal.
- But three medians being equal implies all three sides are equal.
10. How does the median divide a triangle?
A median divides a triangle into two smaller triangles of equal area.
- This is because it connects a vertex to the midpoint of the opposite side
- Each smaller triangle shares the same height and base (half of the full base)
