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Equal Angles Subtend Equal Sides

P and O are...

Question

$P$ and $O$ are the mid points of the sides AB and BC respectively of the triangle ABC, right-angled at B, then :

{A Q}^{2} + {C P}^{2} = {A C}^{2}

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{A Q}^{2} + {C P}^{2} = {\frac{4}{5} A C}^{2}

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{A Q}^{2} + {C P}^{2} = {\frac{5}{4} A C}^{2}

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{A Q}^{2} + {C P}^{2} = {\frac{3}{5} A C}^{2}

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Solution

The correct option is C ${A Q}^{2} + {C P}^{2} = {\frac{5}{4} A C}^{2}$
Construction: Join PC and AQ as shown in the figure.
Consider $Δ A B Q, {A Q}^{2} = A B^{2} + {B Q}^{2} ⟹ {A B}^{2} = A Q^{2} - {B Q}^{2}$
Consider $Δ P B C, {P C}^{2} = P B^{2} + {B C}^{2} ⟹ {B C}^{2} = P C^{2} - {P B}^{2}$
We know that BQ = $\frac{1}{2} B C$ and PB = . $\frac{1}{2} A B$ .
Consider $Δ A B C, {A C}^{2} = A B^{2} + {B C}^{2} {A C}^{2} = A Q^{2} - B Q^{2} + {P C}^{2} - B C^{2} {A C}^{2} = A Q^{2} - \frac{1}{4} B C^{2} + {P C}^{2} - \frac{1}{4} A B^{2} {A C}^{2} = A Q^{2} + {P C}^{2} - \frac{1}{4} (B C^{2} + A B^{2}) {A C}^{2} = A Q^{2} + {P C}^{2} - \frac{1}{4} (A C^{2}) ⟹ \frac{5}{4} (A C^{2}) = A Q^{2} + {P C}^{2}$

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Similar questions

Q. In a right triangle ABC right-angled at B, if P and Q are points on the sides AB and AC respectively, then

(a) AQ² + CP² = 2(AC²+ PQ²)
(b) 2(AQ² + CP²) = AC² + PQ²
(c) AQ²+ CP² = AC² + PQ²
(d)

AQ + CP = \frac{1}{2} (AC + PQ)

Q. In the given figure,

O P, O Q

and

O R

are drawn perpendiculars to the sides

B C, C A

and

A B

respectively of triangle

A B C

. Prove that:

{A R}^{2} + {B P}^{2} + {C Q}^{2} = {A Q}^{2} + {C P}^{2} + {B R}^{2}

Q. In the given figure:

O P, O Q

and

O R

are drawn perpendicular to the sides

A B C, C A

and

A B

respectively of triangle

A B C

. Prove that

{A F}^{2} + {B P}^{2} + {C Q}^{2} = {A Q}^{2} + {C P}^{2} + {B R}^{2}

P and Q are points on sides CA and CB respectively of Δ ABC right angled at C. Prove that ${A Q}^{2}$ + ${B P}^{2}$ = ${A B}^{2}$ + ${P Q}^{2}$ [4 MARKS]

Q. If P and Q are points on sides CA and CB respectively of ΔABC right angled at C, then

{A Q}^{2}

{B P}^{2} = ?

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P and O are the mid points of the sides AB and BC respectively of the triangle ABC, right-angled at B, then :

$P$ and $O$ are the mid points of the sides AB and BC respectively of the triangle ABC, right-angled at B, then :