In $\triangle ABC$, $\angle B = 90^\circ$. $BD \perp AC$. If $AD = 4$ cm and $CD = 5$ cm, find the length of $BD$.

Answer: $2\sqrt{5}$ cm. Using Geometric Mean property: $BD^2 = AD \times CD$. $BD = \sqrt{4 \times 5} = \sqrt{20} = 2\sqrt{5}$ cm.

The centroid of a triangle divides each median in the ratio:

Answer: 2:1. Standard geometric property: The centroid always divides medians in 2:1 ratio from the vertex.

Comprehensive guide to triangle types, centers (Centroid/Incenter), Similarity vs Congruence, and the Midpoint theorem.

Centroid (G): Formed by Medians. Divides median in 2:1.
Incenter (I): Formed by Angle Bisectors. Center of Incircle, equidistant from sides.
Circumcenter (O): Formed by Perpendicular Bisectors. Equidistant from vertices.
Orthocenter (H): Formed by Altitudes (Heights).

If $\triangle ABC \sim \triangle PQR$ , then:

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