If the length of the bases of a trapezoid are 5 and 15 cm. And length of diagonals are 12 cm and 16 cm. Find the area of trapezoid.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should remember that you may evaluate the area of trapezoid using the following formula such that:

`A = (1/2)(b_1 + b_2)*h`

Notice that `b_1`  and `b_2`  represent the lengths of the bases and h represents the height of trapezoid.

The problem provides the lengths of bases, but you need to find the length of the height.

Notice that the bases are perependicular to the height.

You should notice that the two heights are equal.

You should come up with the following notations such that:

`AD ` represents the smaller base

`BC`  represents the larger base

`AD||BC`

`AC, BD`  diagonals

`AE, DF`  heights

`AE = DF`

`BE = x ; CF = y`

Using the Pythagora's theorem in triangles AEC and DFB (`hatE = hatF = 90^o` ) yields:

`AC^2 - (BC - x)^2 = BD^2 - (BC - y)^2`

`16^2 - (15 - x)^2 = 12^2 - (15 - y)^2`

`16^2 - 12^2 = (15 - x)^2 - (15 - y)^2`

Converting the differences of squares into products yields:

`(16 - 12)(16 + 12) = (15 - x - 15 + y)(15 - x + 15 - y)`

`4*28 = (y-x)(30 - (x+y))`

You should also use the equation that relates the lengths of bases and the unknown lengths x and y such that:

`BC - AD = x + y`

`15 - 5 = x + y => 10 = x + y`

Notice that you may substitute `10`  for `x + y`  in the first relation such that:

`4*28 = (y-x)(30 - 10) => y - x = 4*28/20 => y - x = 28/5`

You need to form the system of equations such that:

`{(x + y = 10) , (-x + y = 28/5):} => 2y = 10 + 28/5`

`y = 5 + 14/5 => y = (25+14)/5 => y = 39/5`

`x = 10 - 39/5 => x = (50-39)/5 =>x = 11/5`

You may evaluate now the height of trapezoid such that:

`h^2 = AC^2 - (BC - x)^2 `

`h = sqrt(16^2 - (15 - x)^2)`

`h = sqrt(256 - (15 - 11/5)^2) => h = sqrt(256 - 4096/25)`

`h = sqrt((6400 - 4096)/25) => h = sqrt(2304/25) => h = 48/5`

You need to evaluate the area of trapezoid such that:

`A = (5+15)(48/10) => A = 2*48 => A = 96 cm^2`

Hence, evaluating the area of trapezoid, under the given conditions, yields `A = 96 cm^2` .

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