the area of a rhombus is 48 cm² and its perimeter is 20 cm .what is its height ?
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The area of a rhombus is 48cm square and its perimeter is 20cm. What is its height?
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The area of a rhombus is 48 cm square and its perimeter is 20cm. What is its height?
The sides of the rhombus is 20/4 = 5 cm. The hypotenuse of the RAT is 5 cm.
Area of each RAT =48/4 = 12 sq cm hence the height of each RAT on the hypotenuse as the base = 12/5 = 2.4 cm.
Consider one of the RAT, say ABC, right angle at B. From B drop BD (= 2.4 cm) as a perpendicular on AC which is the hypotenuse and 5 cm long.
Let AD = x and DC = (5-x), ABD and BCD are similar triangles. Thus AD/BD = BD/CD or
x/BD = BD/(5-x), or BD^2 = x(5-x) = 2.4^2 = 5.76
x^2 - 5x + 5.76 = 0, or
x =[5 +(5^2 - 4*5.76)^0.5]/2
= [5 + 1.4]/2 = 6.4/2 = 3.2 cm= AD
In RAT, ABD. AB^2 = AD^2 + BD^2 = 3.2^2 + 2.4^2 = 10.24 +5.76 = 16 or AB = 4 cm.
Area of ABC = 48/4 = 12 sq cm. So if AB = 4 cm, BC should be 12x2/4 = 6 cm
The diagonals of the rhombus are therefore 8 cm and 12 cm which give the area of the rhombus as 8*12/2 = 48 sq cm. Correct.
The rhombus cannot have a height as mentioned in the question. The height on the hypotenuse of the four RATs is 2.4 cm.