Question

ABC is an isosceles right angled triangle, right angled C .prove that AB²=2AC²

Answer 1

HEYA!!
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⬆⬆Refer to the Attachment for the figures ⬆⬆
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We have

AC = BC { triangle is isosceles}........................(1)

Also by Pythagoras theorem , we have


AB^2 = AC^2 + BC^2

AB^2 = 2 AC^2 { putting BC=AC from (1) }


Thus proved !!

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Answer 2

DIAGRAM:

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$\setlength{\unitlength}{1.4 cm}\begin{picture}(0,0) \thicklines\qbezier(0,0)(0,0)(3 ,0)\qbezier(0,0)(0,0)(0 ,3)\qbezier(3,0)(3,0)(0 ,3) \put( - 0.5, - 0.3){ \bf C }\put(2.9, - 0.3){ \bf B }\put( - 0.5, 3){ \bf A } \put( - 0.1 ,0.4){ \line(1 ,0){0.4}}\put( 0.3 ,0.4){ \line(0 , - 1){0.4}} \put( 0.3,0.45){ \bf {90}^{ \circ} }\end{picture}$

⠀⠀⠀⠀⠀⠀ ⠀━━━━━━━━━━━━━━━━━━━━━━━━━━

⠀☯ Since ∆ABC is an Isosceles right triangle right - angled at C.

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Therefore,

• AC = BC

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☯ Now, Using Pythagoras Theorem

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$\qquad\star\;\sf H^2 = P^2 + B^2$

$:\implies\sf AB^2 = AC^2 + BC^2$

$:\implies\sf AB^2 = AC^2 + AC^2\qquad\qquad\bigg\lgroup\bf \because\;BC = AC\bigg\rgroup$

$:\implies{\underline{\boxed{\sf{AB = 2AC^2}}}}\;\bigstar$

$\dag\;{\underline{\bf{Hence\;Proved!}}}$

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$\qquad\quad\boxed{\underline{\underline{\bigstar \:\bf\:More\:to\:know\:\bigstar}}}$

★ Isosceles Right triangle -

• In an isosceles right angled triangle, the equal sides make the right angle (90°). They have the ratio of equality 1:1.

★ Pythagoras Theorem -

• Pythagoras theorem states that "In a right-angled triangle, the square of the hypotenuse is equal to the sum of squares of the other two sides i.e base and perpendicular".

• The sides of this triangle have been named as Perpendicular, Base and Hypotenuse.

• Here, the hypotenuse is the longest side, as it is opposite to the angle 90°.