Apollonius Theorem

Seg $\mathrm{PM}$ is a median of $\mathrm{\Delta PQR}$. If $\mathrm{PQ}=40, \mathrm{PR}=42$ and $\mathrm{PM}=29,$ find $\mathrm{QR}$.

$\mathrm{SegAM}$ is a median of $△\mathrm{ABC}$. If $\mathrm{AB}=22, \mathrm{AC}=34, \mathrm{BC}=24.$ Find $\mathrm{AM}$.

In the figure, $\mathrm{\Delta }$ $\mathrm{PQR}$ is an equilateral triangle. Point $\mathrm{S}$ is on $seg\mathrm{QR}$ such that $\mathrm{QS}=\frac{1}{3}\mathrm{QR}$. Prove that $: 9{\mathrm{PS}}^2=7{\mathrm{PQ}}^2$. 

In $\mathrm{\Delta ABC}, \angle \mathrm{BAC}=90°$ $\mathrm{seg} \mathrm{BL}$ and $\mathrm{seg} \mathrm{CM}$ are medians of $△\mathrm{ABC}$. Then prove that $4\left({\mathrm{BL}}^2+{\mathrm{CM}}^2\right)=5{\mathrm{BC}}^2$

From the information given in the, prove that $\mathrm{PM}=\mathrm{PN}=\sqrt{3}\times \mathrm{a}$

In the given figure point $\mathrm{T}$  is in the interior of rectangle $\mathrm{PQRS}$. Prove that, ${\mathrm{TS}}^2+{\mathrm{TQ}}^2={\mathrm{TP}}^2+{\mathrm{TR}}^2$

(As shown in the figure, draw

Seg $\mathrm{AB}\Vert$ Side $SR$ and  $A-T-B$)

In $\mathrm{\Delta ABC}$, point $\mathrm{M}$ is the midpoint of side $\mathrm{BC}$. If ${\mathrm{AB}}^2+{\mathrm{AC}}^2=290{\mathrm{cm}}^2$, $\mathrm{AM}=8\mathrm{cm},$  find $\mathrm{BC}.$

In the given figure seg PS is the median of $\mathrm{\Delta PQR}$ and $\mathrm{PT}\perp \mathrm{QR}$ Prove that,

${\mathrm{PQ}}^2={\mathrm{PS}}^2-\mathrm{QR}\times \mathrm{ST}+{\left(\frac{\mathrm{QR}}{2}\right)}^2$

In the given figure seg $\mathrm{PS}$ is the median of $\Delta \mathrm{PQR}$ and $\mathrm{PT}\perp \mathrm{QR}$
Prove that, ${\mathrm{PR}}^2={\mathrm{PS}}^2+\mathrm{QR}\times \mathrm{ST}+{\left(\frac{\mathrm{QR}}{2}\right)}^2$

In $\mathrm{\Delta ABC},\mathrm{AB}=10,\mathrm{AC}=7,\mathrm{BC}=9$ then find the length of the median drawn from point $C$ to side $A B$

In $△\mathrm{PQR}$, point $\mathrm{S}$ is the midpoint of side $\mathrm{QR}$. If $\mathrm{PQ}=11,\mathrm{PR}=17,\mathrm{PS}=13$, find $\mathrm{QR}$