From the information given in the adjoining figure, Prove that: PM = P

1.	From the information given in the adjoining figure, Prove that: PM = PN = √3 x a, where QR = a.
Answer» Proof: In ∆PMR, QM = QR = a ---- [Given] \(\therefore\) Q is midpoint of seg MR. \(\therefore\) seg PQ is the median \(\therefore\) PM² + PR² = 2PQ² + 2QM² ---- [By Apollonius theorem] \(\therefore\) PM² + a² = 2a² + 2a² ---- [Substituting the given values] \(\therefore\) PM² + a² = 4a² \(\therefore\) PM² = 4a² - a² \(\therefore\) PM² = 3a² \(\therefore\) PM = √3 a ---- [Taking square root on both sides] Similarly, we can prove PN = √3 a \(\therefore\) PM = PN = √3 a

From the information given in the adjoining figure, Prove that: PM = PN = √3 x a, where QR = a.

Answer»

Proof:

In ∆PMR,

QM = QR = a ---- [Given]

\(\therefore\) Q is midpoint of seg MR.

\(\therefore\) seg PQ is the median

\(\therefore\) PM² + PR² = 2PQ² + 2QM² ---- [By Apollonius theorem]

\(\therefore\) PM² + a² = 2a² + 2a² ---- [Substituting the given values]

\(\therefore\) PM² + a² = 4a²

\(\therefore\) PM² = 4a² - a²

\(\therefore\) PM² = 3a²

\(\therefore\) PM = √3 a ---- [Taking square root on both sides]

Similarly, we can prove

PN = √3 a

\(\therefore\) PM = PN = √3 a

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From the information given in the adjoining figure, Prove that: PM = PN = √3 x a, where QR = a.

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