Q. The sum of the two shortest sides of a right-angles triangle is 18cm.Calculate
a.the least possible length of the hypotenuse
b.the greatest possible area of the triangle
shortest side ---- x
longer side = 18-x
h^2 = x^2 + (18-x)^2
= 2x^2 - 36x + 324
d(h^2)/dx = 4x - 36
= 0 for a max/min ( for h to be a minimum, h^2 would have to be a minimum) x = 9
when x = 9 , h^2 = 2(81) - 36(9) + 324 = 162
h = √162 or appr 12.728
area = (1/2)(x)(18-x)
= 9x - x^2/2
d(area)/dx = 9 - x/4 = 0 for maximum area
etc.
To solve this problem, we need to use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Let's solve part (a) first:
a. The least possible length of the hypotenuse:
Since we are given that the sum of the two shortest sides is 18 cm, let's assume the two shortest sides are x and y. According to the question, x + y = 18 cm.
Now, let's apply the Pythagorean theorem:
x^2 + y^2 = hypotenuse^2
Since we want to find the least possible length of the hypotenuse, we should find the minimum value of the hypotenuse. This happens when the two shortest sides are equal (x = y).
Substituting x = y and x + y = 18:
2x = 18
x = 9 cm
So, the shortest side is 9 cm, and by symmetry, the other shortest side is also 9 cm.
Now, let's find the hypotenuse:
hypotenuse^2 = 9^2 + 9^2
hypotenuse^2 = 162
hypotenuse ≈ 12.73 cm
Therefore, the least possible length of the hypotenuse is approximately 12.73 cm.
Now let's solve part (b):
b. The greatest possible area of the triangle:
The area of a right-angled triangle can be calculated using the formula: A = (base * height) / 2.
In this case, the base and height are the two shortest sides of the triangle (9 cm each).
The area is given by:
A = (9 * 9) / 2
A = 81 / 2
A ≈ 40.5 cm^2
Thus, the greatest possible area of the triangle is approximately 40.5 cm^2.