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E perimeter of a rectangle is 28 meters, and the diagonal is 10 meters,

what is the area?​

User NelsonGon
by
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1 Answer

6 votes

Answer:

The rectangle's area is 48 square meters

Explanation:

Recall that the perimeter of a rectangle of base b and height h is given by the formula:

Perimeter = 2 b + 2 h

we know that the perimeter is 28 meters, then we can create our first equation;

2 b + 2 h = 28

which means:

2 (b + h) = 28

b + h = 28/2

b + h = 14

the tell us that the diagonal is 10 meters, so we use the Pythagorean theorem to write a second equation using the rectangle's base, height, and diagonal (which form in between the three a right angle triangle where the hypotenuse is the rectangle's diagonal:


h^2+b^2=10^2=100

So, we can use the equation : b + h = 14 to write one variable in terms of the other one and use it as substitution in the second (quadratic) equation:

h = 14 - b

then:


h^2+b^2=100\\(14-b)^2+b^2=100\\14^2-28\,b+b^2+b^2=100\\2\,b^2-28\,b+196-100=0\\2\,b^2-28\,b+96=0\\b^2-14\,b+48=0

which we have reduced at the end by dividing both sides by 2.

we can use factoring to solve these equation;


b^2-14\,b+48=0\\b^2-6\,b-8\,b+48=0\\b\,(b-6)-8\,(b-6)=0\\(b-6)\,(b-8)=0

Se we find two possible solutions: b = 6 m or b = 8 m

If we call b = 8 m, then the height becomes h = 14 - (8) = 6 m

and viceversa.

So a rectangle with such dimensions will render an area that equals :

Area = b x h = 8 x 6 = 48 square meters.

User Emre Karahan
by
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