Question

X + 2

Diagram NOT
accurately drawn
x-5
x + 6
The diagram shows a trapezium.
The lengths of three of the sides of the trapezium are x - 5, x + 2 and x + 6.
All measurements are given in centimetres.
The area of the trapezium is 36 cm.
(a) Show that x^2- x - 56 = 0

-look at the picture (a)

Answer 1

By using the trapezium area formula, we established that the area in terms of x is (x+4)(x-5), which simplifies to x^2 - x - 20. Setting this equal to the given area of 36 cm^2 and rearranging, we derive the equation x^2 - x - 56 = 0.

Step-by-step explanation:

To demonstrate that x2 - x - 56 = 0, we must use the properties of a trapezium and the given area. The area A of a trapezium can be calculated with the formula A = \((a+b)/2\) * h, where a and b are the lengths of the parallel sides and h is the height. In this case, one pair of parallel sides is x+2 and x+6, and the area is given as 36 cm2. If we let the height h of the trapezium be x - 5, then the area is:

A = \((x+2+x+6)/2\) * (x-5)

36 = \((2x+8)/2\) * (x-5)

36 = (x+4)(x-5)

36 = x2 - 5x + 4x - 20

36 = x2 - x - 20

Adding 20 to both sides:

56 = x2 - x

Therefore, the equation becomes:

x2 - x - 56 = 0

This matches the equation we needed to show, thus completing the proof.

Answer 2

see explanation

Step-by-step explanation:

The area (A) of a trapezium is calculated as

A =
`(1)/(2)` h (a + b)

where a, b are the measures of the parallel sides and h the perpendicular distance between them.

here h = x - 5, a = x + 6 and b = x + 2, thus

A =
`(1)/(2)`(x - 5)(x + 6 + x + 2) = 36

Multiply both sides by 2 to clear the fraction

(x - 5)(2x + 8) = 72 ← expand left side

2x² - 2x - 40 = 72 ← divide through by 2

x² - x - 20 = 36 ( subtract 36 from both sides )

x² - x - 56 = 0 ← as required