Question

abc is a right angled triangle such that ab =ac and bisctor of angle c intersects the side ab at d prove that ac+ad=bc
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Answer 1

Answer:

so you get something like this

let AC = x

then BC = 2–√ x

as AD and DB will be in ratio of AC and BC by angle bisector theorem

so let AD = y and so DB = 2–√ y

but AD + DB = AB = AC

so y+ 2–√ y = x

y( 2–√ + 1) = x

so y = x2–√+1

on rationalising the denominator it results in

y = ( 2–√ - 1) x

add x on both sides

y + x= ( 2–√ - 1) x + x

AD + AC = ( 2–√ - 1 + 1) x

AD + AC = ( 2–√ ) x

so AC + AD = BC

Alternatively it becomes very easy if you know how to calculate

trignometric ratios for 22.5 deg

then AD/AC is just tan(22.5deg) and you can very easily get things done from there.

// hope it helps

Step-by-step explanation:

so you get something like this

let AC = x

then BC = 2–√ x

as AD and DB will be in ratio of AC and BC by angle bisector theorem

so let AD = y and so DB = 2–√ y

but AD + DB = AB = AC

so y+ 2–√ y = x

y( 2–√ + 1) = x

so y = x2–√+1

on rationalising the denominator it results in

y = ( 2–√ - 1) x

add x on both sides

y + x= ( 2–√ - 1) x + x

AD + AC = ( 2–√ - 1 + 1) x

AD + AC = ( 2–√ ) x

so AC + AD = BC

Alternatively it becomes very easy if you know how to calculate

trignometric ratios for 22.5 deg

then AD/AC is just tan(22.5deg) and you can very easily get things done from there.

// hope it helps