# In a triangle ABC, N is appointed on AC such that BN is perpendicular to AC. If $B{N^2} = AN \cdot NC$, prove that $\angle B = {90^0}$.

Last updated date: 18th Mar 2023

•

Total views: 307.8k

•

Views today: 6.86k

Answer

Verified

307.8k+ views

Hint: - Use Pythagoras Theorem, $\left[ {{{\left( {{\text{Hypotenuse}}} \right)}^2} = {{\left( {{\text{Perpendicular}}} \right)}^2} + {{\left( {{\text{Base}}} \right)}^2}} \right]$

Given:

BN is perpendicular to AC

\[\therefore \angle BNC = \angle BNA = 90\]

And it is also given that $B{N^2} = AN \cdot NC............\left( 1 \right)$

Apply Pythagoras Theorem in $\Delta BNC$

$\therefore {\left( {BC} \right)^2} = {\left( {BN} \right)^2} + {\left( {NC} \right)^2}$

From equation 1

${\left( {BC} \right)^2} = \left( {AN \times NC} \right) + {\left( {NC} \right)^2}.................\left( 2 \right)$

Apply Pythagoras Theorem in $\Delta BNA$

$\therefore {\left( {BA} \right)^2} = {\left( {BN} \right)^2} + {\left( {NA} \right)^2}$

From equation 1

$\therefore {\left( {BA} \right)^2} = \left( {AN \times NC} \right) + {\left( {NA} \right)^2}.........\left( 3 \right)$

Add equations 2 and 3

$

{\left( {BC} \right)^2} + {\left( {BA} \right)^2} = \left( {AN \times NC} \right) + {\left( {NC} \right)^2} + \left( {AN \times NC} \right) + {\left( {NA} \right)^2} \\

\therefore {\left( {BC} \right)^2} + {\left( {BA} \right)^2} = {\left( {NC} \right)^2} + {\left( {NA} \right)^2} + 2\left( {AN \times NC} \right) \\

$

In above equation R.H.S is the formula of ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$

$\therefore {\left( {BC} \right)^2} + {\left( {BA} \right)^2} = {\left( {AN + NC} \right)^2}$

From figure $AN + NC = AC$

$\therefore {\left( {BC} \right)^2} + {\left( {BA} \right)^2} = {\left( {AC} \right)^2}$

Which is the property of Pythagoras Theorem.

Where AC is hypotenuse, AB and BC are perpendicular to each other at B.

$\therefore \angle B = {90^0}$

Hence Proved.

Note: - whenever we face such types of problems first draw the pictorial representation of the given problem, then apply Pythagoras Theorem which is stated above, then according to given condition simplify the problem, then we will get the required answer.

Given:

BN is perpendicular to AC

\[\therefore \angle BNC = \angle BNA = 90\]

And it is also given that $B{N^2} = AN \cdot NC............\left( 1 \right)$

Apply Pythagoras Theorem in $\Delta BNC$

$\therefore {\left( {BC} \right)^2} = {\left( {BN} \right)^2} + {\left( {NC} \right)^2}$

From equation 1

${\left( {BC} \right)^2} = \left( {AN \times NC} \right) + {\left( {NC} \right)^2}.................\left( 2 \right)$

Apply Pythagoras Theorem in $\Delta BNA$

$\therefore {\left( {BA} \right)^2} = {\left( {BN} \right)^2} + {\left( {NA} \right)^2}$

From equation 1

$\therefore {\left( {BA} \right)^2} = \left( {AN \times NC} \right) + {\left( {NA} \right)^2}.........\left( 3 \right)$

Add equations 2 and 3

$

{\left( {BC} \right)^2} + {\left( {BA} \right)^2} = \left( {AN \times NC} \right) + {\left( {NC} \right)^2} + \left( {AN \times NC} \right) + {\left( {NA} \right)^2} \\

\therefore {\left( {BC} \right)^2} + {\left( {BA} \right)^2} = {\left( {NC} \right)^2} + {\left( {NA} \right)^2} + 2\left( {AN \times NC} \right) \\

$

In above equation R.H.S is the formula of ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$

$\therefore {\left( {BC} \right)^2} + {\left( {BA} \right)^2} = {\left( {AN + NC} \right)^2}$

From figure $AN + NC = AC$

$\therefore {\left( {BC} \right)^2} + {\left( {BA} \right)^2} = {\left( {AC} \right)^2}$

Which is the property of Pythagoras Theorem.

Where AC is hypotenuse, AB and BC are perpendicular to each other at B.

$\therefore \angle B = {90^0}$

Hence Proved.

Note: - whenever we face such types of problems first draw the pictorial representation of the given problem, then apply Pythagoras Theorem which is stated above, then according to given condition simplify the problem, then we will get the required answer.

Recently Updated Pages

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

A Short Paragraph on our Country India