prove that the area of the equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonal.
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prove that the area of the equilateral triangle described on one side of a square is equal to half the area of the equilateral triangle described on one of its diagonal.
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Solution:
Areas of Similar Triangles NCERT Solutions Tenth Grade
Given: ABCD is a square, AEB is an equilateral triangle described on the side of the square, DBF is an equilateral triangle described on diagonal BD of square.
To Prove: ar(△DBF)ar(△AEB)=21
Proof: Any two equilateral triangles are similar because all angles are of 60 degrees.
Therefore, by AAA similarity criterion, △DBF ~ △AEB
ar(△DBF)ar(△AEB)=DB2AB2 (1)
{The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides}
But, we have DB=2–√AB {Diagonal of square is 2–√ times of its side} (2)
Putting equation (2) in equation (1), we get
ar(△DBF)ar(△AEB)=(2√AB)2AB2=2AB2AB2 = 2
Therefore, area of equilateral triangle described on one side os square is equal to half the area of the equilateral triangle described on one of its diagonals.
Hence Proved