Q. 224.3( 23 Votes )
AD is an altitude of an equilateral triangle ABC. On AD as base, another equilateral triangle ADE is constructed. Prove that Area () : Area () = 3 : 4.
To prove: Area () : Area () = 3 : 4.
Construct the figure according to the conditions given.
ABC is an equilateral triangleLet one side AB be 2X
Since in equilateral triangle all the sides are of equal length.
⇒ AB=BC=AC= 2X
Since perpendicular bisects the given side into two equal parts,
Now, In ADB
By Pythagoras theorem,
AB2 = AD2 + BD2
AD2=AB2 - BD2
AD2 = (2x)2-(x)2
ABC and ADE both are equilateral trianglesSince, all the angles of the equilateral triangle are of 60°.
∴ABCADE [By AA similarity]
By the theorem which states that the areas of two similar triangles are in the ratio
of the squares of the any two corresponding sides.
Hence,Area () : Area () = 3 : 4
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