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Question
Equilateral triangles are drawn on the sides of a right angled triangle. Show that the area of the triangle on the hypotenuse is equal to the sum of the areas of triangles on the other two sides.
Solution
Given A right angled triangle ABC with right angle at B. Equilateral triangles PAB, QBC and RAC are described on sides AB, BC and CA respectively.
To Prove.
Area (ΔPAB) + Area (ΔQBC) = Area (ΔRAC).
Proof. Since, triangles PAB, QBC and RAC are equilateral. Therefore they are equiangular and hence similar.
∴ `"area (ΔPAB)"/"area (ΔRAC)" + "area (ΔQBC)"/"area (ΔRAC)"`
= `"AB"^2/"AC"^2 + "BC"^2/"AC"^2`
= `("AB"^2 + "BC"^2)/("AC"^2)`
= `"AC"^2/"AC"^2` = 1
`[∵ "ΔABC is a right angled triangle with" ∠"B" = 90°
∴ "AC"^2 = "AB"^2 + "BC"^2 ]`
⇒ `"area (ΔPAB) + area (ΔQBC)"/"area (ΔRAC)"` = 1
⇒ area (ΔPAB) + area (ΔQBC)
= area (ΔRAC).
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