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Quadrilaterals - A Surprising Property of Medians: The Centroid Theorem

Grade 9CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has exactly three medians, which are concurrent (they meet at a single point).

Triangle ABC with median AD where D is the midpoint of BC.
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The point of concurrency of the three medians is called the Centroid (often denoted as GG). A key property of the centroid is that it divides each median in the ratio 2:12:1, with the longer segment being the one adjacent to the vertex.

Triangle with all three medians intersecting at the centroid G.
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In any quadrilateral, the segments joining the midpoints of opposite sides and the segment joining the midpoints of the diagonals all bisect each other at a single point. This point is the centroid of the quadrilateral's vertices.

Quadrilateral with lines connecting midpoints of opposite sides.

📐Formulae

Ratio of Centroid Division=AG:GD=2:1\text{Ratio of Centroid Division} = AG:GD = 2:1

Length of Median ma=122b2+2c2−a2\text{Length of Median } m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}

G⃗=A⃗+B⃗+C⃗3\vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3}

Area(△AGB)=Area(△BGC)=Area(△CGA)=13Area(△ABC)\text{Area}(\triangle AGB) = \text{Area}(\triangle BGC) = \text{Area}(\triangle CGA) = \frac{1}{3} \text{Area}(\triangle ABC)

💡Examples

Problem 1:

In △ABC\triangle ABC, ADAD is a median and GG is the centroid. If the length of AD=12 cmAD = 12 \text{ cm}, find the lengths of AGAG and GDGD.

Median AD divided by centroid G.

Solution:

  1. According to the Centroid Theorem, the centroid GG divides the median ADAD in the ratio 2:12:1.
  2. This means AG=22+1×ADAG = \frac{2}{2+1} \times AD and GD=12+1×ADGD = \frac{1}{2+1} \times AD.
  3. Calculate AGAG: AG=23×12=8 cmAG = \frac{2}{3} \times 12 = 8 \text{ cm}
  4. Calculate GDGD: GD=13×12=4 cmGD = \frac{1}{3} \times 12 = 4 \text{ cm} Final Answer: AG=8 cmAG = 8 \text{ cm} and GD=4 cmGD = 4 \text{ cm}.

Explanation:

The theorem states that the distance from the vertex to the centroid is twice the distance from the centroid to the midpoint of the opposite side.

Problem 2:

In △PQR\triangle PQR, PTPT is a median. If GT=5 unitsGT = 5 \text{ units} where GG is the centroid, find the length of the segment PGPG and the total length of the median PTPT.

Triangle PQR with median PT and centroid G.

Solution:

  1. We know from the property of medians that PG:GT=2:1PG:GT = 2:1.
  2. Given GT=5GT = 5, we can find PGPG: PG=2×GT=2×5=10 unitsPG = 2 \times GT = 2 \times 5 = 10 \text{ units}
  3. The total length of the median PTPT is the sum of PGPG and GTGT: PT=PG+GT=10+5=15 unitsPT = PG + GT = 10 + 5 = 15 \text{ units} Final Answer: PG=10 unitsPG = 10 \text{ units} and PT=15 unitsPT = 15 \text{ units}.

Explanation:

By applying the ratio 2:12:1, if the smaller part of the median (from centroid to side) is known, the larger part is double that value.