Question
Mathematics Question on Triangles
An equilateral triangle of side 43 cm formed out of a sheet is converted into a rectangle such that there is no loss of the area of the triangle. Then the least perimeter of the rectangle (in cm) will be:
23
43
12
83
83
Solution
Calculate the area of the equilateral triangle. The formula for the area of an equilateral triangle with side a is:
Area =43⋅a2
Substitute a=43:
Area =43⋅(43)2=43⋅48=123cm2
Dimensions of the rectangle. The rectangle has the same area as the triangle. Let the dimensions of the rectangle be l (length) and b (breadth). Then:
l⋅b=123
For the rectangle to have the least perimeter, it should be as close to a square as possible (to minimize l+b). Hence, let:
l=b
Then:
l2=123⟹l=123=2412≈23cm
Thus, the dimensions are:
l=b=23cm
Perimeter of the rectangle. The perimeter of a rectangle is given by:
Perimeter =2(l+b)
Substitute l=b=23:
Perimeter =2(23+23)=2⋅43=83cm
Final Answer: The least perimeter of the rectangle is:
83