in the figure, PQR is a triangle in which QM perpendicular PR and PR2 -PQ 2 =QR 2 ,prove that - Maths - Triangles

Question

in the figure, PQR is a triangle in which QM perpendicular PR
and PR2 -PQ 2 =QR 2 ,prove that
QM2=PMxMR

Pronay · Accepted Answer

Given: ${PR}^{2} - {PQ}^{2} = {QR}^{2} \Rightarrow$ ΔPQR is right angle triangle right angle at Q. QM is perpendicular to PR

To show:

${QM}^{2} = PM \times M R$

By given condition

$\Rightarrow {PR}^{2} = {PQ}^{2} + {QR}^{2} ...... (1)$

By Pythagoras theorem in ΔQMP

$\Rightarrow {QP}^{2} = {QM}^{2} + {PM}^{2} ...... (2)$

By Pythagoras theorem in ΔQMR

$\Rightarrow {QR}^{2} = {QM}^{2} + {MR}^{2} ...... (3)$

$(1) \Rightarrow {PR}^{2} = {PQ}^{2} + {QR}^{2} \Rightarrow {PR}^{2} = {QM}^{2} + {PM}^{2} + {QM}^{2} + {M R}^{2} (by (1) and (2)) \Rightarrow {PR}^{2} = 2 {QM}^{2} + {PM}^{2} + {MR}^{2} + 2 PM \times M R - 2PM \times MR \Rightarrow {PR}^{2} = 2 {QM}^{2} + {(PM+MR)}^{2} - 2 PM \times MR \Rightarrow {PR}^{2} = 2 {QM}^{2} + {PR}^{2} - 2 \times PM \times M R \Rightarrow {QM}^{2} = PM \times MR$

in the figure, PQR is a triangle in which QM perpendicular PR and PR2 -PQ 2 =QR 2 ,prove that QM2=PMxMR .

in the figure, PQR is a triangle in which QM perpendicular PR and PR² -PQ ²=QR² ,prove that QM²=PMxMR .