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Property of Median Drawn on Hypotenuse of the Right Triangle

In the given ...

Question

In the given figure seg PS is the median of ∆PQR and PT ⊥ QR. Prove that,

(1) ${PR}^{2} = {PS}^{2} + QR \times ST + {(\frac{QR}{2})}^{2}$

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

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Solution

According to Pythagoras theorem, in ∆PTQ

${PQ}^{2} = {PT}^{2} + {QT}^{2} . . . (1)$

In ∆PTS

${PS}^{2} = {PT}^{2} + {TS}^{2} . . . (2)$

In ∆PTR

${PR}^{2} = {PT}^{2} + {RT}^{2} . . . (3)$

In ∆PQR, point S is the midpoint of side QR.

$QS = SR = \frac{1}{2} QR . . . (4)$

${PQ}^{2} + {PR}^{2} = 2 {PS}^{2} + 2 {QS}^{2} (by Apollonius theorem) \Rightarrow {PR}^{2} = 2 {PS}^{2} + 2 {QS}^{2} - {PQ}^{2} \Rightarrow {PR}^{2} = {PS}^{2} + {PS}^{2} + {QS}^{2} + {QS}^{2} - {PQ}^{2} \Rightarrow {PR}^{2} = {PS}^{2} + ({PT}^{2} + {TS}^{2}) + {QS}^{2} + {(\frac{QR}{2})}^{2} - ({PT}^{2} + {QT}^{2}) (From (1), (2) and (4)) \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {PT}^{2} + {TS}^{2} + {QS}^{2} - {PT}^{2} - {QT}^{2} \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} + ({TS}^{2} - {QT}^{2}) \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} + (TS + QT) (TS - QT) \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} + (QS) (TS - QT) \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} + QS \times TS - QS \times QT \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + QS \times TS + {QS}^{2} - QS \times QT \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + QS \times TS + QS (QS - QT) \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + QS \times TS + QS \times TS \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + 2 QS \times TS \Rightarrow {PR}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + QR \times TS$

Hence, ${PR}^{2} = {PS}^{2} + QR \times ST + {(\frac{QR}{2})}^{2}$ .

Now,

${PQ}^{2} + {PR}^{2} = 2 {PS}^{2} + 2 {QS}^{2} (by Apollonius theorem) \Rightarrow {PQ}^{2} = 2 {PS}^{2} + 2 {QS}^{2} - {PR}^{2} \Rightarrow {PQ}^{2} = {PS}^{2} + {PS}^{2} + {QS}^{2} + {QS}^{2} - {PR}^{2} \Rightarrow {PQ}^{2} = {PS}^{2} + ({PT}^{2} + {TS}^{2}) + {QS}^{2} + {(\frac{QR}{2})}^{2} - ({PT}^{2} + {RT}^{2}) (From (2), (3) and (4)) \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {PT}^{2} + {TS}^{2} + {QS}^{2} - {PT}^{2} - {RT}^{2} \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} - ({RT}^{2} - {TS}^{2}) \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} - (TS + RT) (RT - TS) \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} - (TS + RT) (RS) \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} - (TS + RT) (QS) \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} + {QS}^{2} - QS \times TS - QS \times RT \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} - QS \times TS + {QS}^{2} - QS \times RT \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} - QS \times TS - QS (RT - QS) \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} - QS \times TS - QS (RT - SR) \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} - QS \times TS - QS \times TS \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} - 2 QS \times TS \Rightarrow {PQ}^{2} = {PS}^{2} + {(\frac{QR}{2})}^{2} - QR \times TS$

Hence, ${PQ}^{2} = {PS}^{2} - QR \times ST + {(\frac{QR}{2})}^{2}$ .

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