Question

prove that the area of an equilateral triangle described on one side of the square is equal to the half the area of the equilateral triangle described on one of its diagonal ​

Answer 1

Answer:

let side of first triangle be s

let side of second triangle be S

area of triangle 1÷ by area of triangle 2=. √3/4s^2/

Answer 2

_________________ANSWER_____________

Given: ΔABC and ΔPQC are two equilateral triangle

ΔPQC described on the diagonal PC

And ΔABC described on side AC

PACN is square

To proof : ar(ΔABC) = 1/2ar(ΔPQC)

proof : since ΔABC and ΔPQC are two equilateral triangle. Thus all angles are of 60°

⇒ΔABC ∽ ΔPQC [by AA similarity]

And

In ΔAPC

PC²=AP²+AC² (by Pythagoras theorem)

PC² =AC² +AC² (AP=AC as PACN is square)

PC² =2AC²

PC =√2AC

⇒AC/CP =1/√2

Now,

By area theorem,

ar(ΔABC)/ar(ΔPQC) =AC²/CP²

=(1/√2)²

ar(ΔABC)/ar(ΔPQC) =1/2

⇒2ar(ΔABC ) =ar(ΔPQC)

⇒ar(ΔABC) =1/2ar(PQC)

HENCE PROVED

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