which ser of lenths will form a triangle with greater area? 6ft,8ft, 10ft or 6ft,8ft,12ft...
Two sides of a triangle with fixed lengths will give the largest area when they form a 90° angle, irrespective of the length of the third side.
Now use Pythagoras theorem to check if the given data fit in the above case.
Proof (it may be beyond what you are doing, but good for reference):
The area of a triangle with side lengths a and b, and included angle C is given by:
Area=a*b*sin(C)/2
and for given a and b, sin(C) is maximum when C=90°.
To determine which set of lengths will form a triangle with a greater area, we can use Heron's formula.
Heron's formula states that the area of a triangle with side lengths a, b, and c can be calculated by using the semi-perimeter, s, which is half the sum of the three side lengths.
The formula for the area, A, is given as:
A = √(s(s-a)(s-b)(s-c))
Now, let's calculate the areas for each set of lengths:
1) For the set of lengths of 6ft, 8ft, and 10ft:
First, calculate the semi-perimeter, s:
s = (6 + 8 + 10) / 2 = 24 / 2 = 12ft
Now, substitute the values into Heron's formula:
A = √(12(12-6)(12-8)(12-10))
= √(12 * 6 * 4 * 2)
= √(576)
= 24ft²
2) For the set of lengths of 6ft, 8ft, and 12ft:
Again, calculate the semi-perimeter, s:
s = (6 + 8 + 12) / 2 = 26 / 2 = 13ft
Now, substitute the values into Heron's formula:
A = √(13(13-6)(13-8)(13-12))
= √(13 * 7 * 5 * 1)
= √(455)
≈ 21.33ft²
Comparing the areas calculated, we can see that the triangle formed by the set of lengths 6ft, 8ft, and 10ft has a greater area of 24ft², while the triangle formed by the set of lengths 6ft, 8ft, and 12ft has an area of approximately 21.33ft².