What is the largest possible area, in square metres, of Sarah's triangular field?

Question

Sarah has been given a field in the shape of a triangle. Two sides of the triangle are exactly 10 metres long.

Answer 1

There's a lot of unnecessary maths above and some is wrong (equilateral triangle, sides = 10, has area 10x5xSqrt(3)/2 = 43.3).

Flipco3108 has it in a nutshell - one 10m side is the base, swinging the other 10m side around one end of base, clearly gives a max. height = 10m when the two are perpendicular.

Area = base x height/2 = 50 sq.m [QED]

(This is clearly a question in logic rather than clever maths)

Answer 2

If the 2 10m sides of the triangle form the apex. Then when the angle between them is 0 degrees ( both sides are touching each other). Then the area fo the field is zero. Similarly if both sides of the triangle are at 180 degrees to each other then they form a straight line. and the area of the field is zero. Between these two ( a vertical line and a horizontal line) is the maximum area of the field. The mid point between 0 degrees and 180 degrees is 90 degrees.. So when the angle at the apex is 90 degrees = max area of field. Area of triangle = 0.5 base * height
Therefore area = (10 * 10)0.5 = 50 m^2 and is isosceles
And Using Pythagoras
length of remaining side = sq root 2 *10 =14.1421m

Answer 3

Area = !/2Base x Height

Base = 1/2 L sin (a/2)
Height = Lcos(a/2)

where L is the given side length = 10 m and a is the angle formed by the 2 equal, given sides.

Thus Area = 1/2 x 1/2 L^2 sin(a/2)cos(a/2) = (1/8) L^2sin(a) , using sin(a )= 2sin(a/2)cos(a/2)

Thus largest area is now obtained by calculus finding the max. of the Area having a as the independent variable i.e. A = f(a)
dA/da = !/8 L^2 cosa = 0 for an extreme.(max or min)==> a = 90º

So it is a right angled triangle of sides 10 each and so hypothenuse = sqrt(10^2 + 10^2 )= 10sqrt(2)

The area is 1/2(10x10) = 50 sq m

Answer 4

As the area of a triangle is base * perpendicular height divided by 2, the largest possible area must have the largest perpendicular height, meaning that it must be a right angled triangle.
This means that the answer is 10*10/2, =50 sq m.

Answer 5

The largest area would be a triangle that fits into a circle where each of the three corners touches the circumference.

This will make the triangle an equalateral, where each side is 10 metres.

Hence, area is 50 sq metres.

Answer 6

this is so easy.

if two sides are the same length, then I assume you have forgotten to tell us that it is a right angled triangle.

thus it is half a square, whose sides are 10m long

10 x 10 = 100m2

half = 50 m2

the formular has been simplified to
Area = 1/2 base x height

however, if it is an Isosceles triangle, and the third side is not the hypotenuses, you need more complex maths, and the angle of the two 10m long sides...

Answer 7

The area of a triangle, in terms of its sides a, b, and c:

a = b = 10
s = (a + b + c ) / 2
Area = sqrt( s(s-a)(s-b)(s-c) )

This area is maximum at c = sqrt(200) = 14.142136
The maximum area is 50 m^2

Answer 8

Area = Half base x perpendicular hight. Therefore 5 x 10 = 50 m2

Answer 9

50

Answer 10

assuming the triangle is a right triangle and both legs are 10 (i'm not sure) then...

Area = 1/2 base x height
A = 1/2 (10) x 10
A = 50