A right triangle with hypotenuse y and equal legs of size x has a perimeter of 36m. Express the area of the triangle solely in terms of the hypotenuse y.
To express the area of the right triangle solely in terms of the hypotenuse y, we need to find a formula that relates the hypotenuse to the area of the triangle.
Let's start by using the Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
In this case, since the legs of the right triangle are equal and their length is x, we can write the Pythagorean theorem as:
x^2 + x^2 = y^2
Simplifying this equation, we get:
2x^2 = y^2
Next, let's find the lengths of the legs in terms of y. Since the perimeter of the triangle is 36m, we can use that information to set up an equation:
x + x + y = 36
Simplifying this equation, we get:
2x + y = 36
Now, solve the second equation for x:
2x = 36 - y
Divide both sides by 2:
x = (36 - y) / 2
Substitute this value for x in the equation 2x^2 = y^2:
2(36 - y)^2 = y^2
Expand and simplify this equation:
2(1296 - 72y + y^2) = y^2
Simplifying further:
2592 - 144y + 2y^2 = y^2
Rearranging this equation to isolate y^2:
2y^2 - 144y + 2592 = y^2
y^2 - 144y + 2592 = 0
Now we have a quadratic equation in terms of y. To find the area of the triangle, we need to compute the y-values that satisfy this equation and take the positive solution since lengths cannot be negative.
Using the quadratic formula, where a = 1, b = -144, and c = 2592:
y = [-(-144) ± √((-144)^2 - 4(1)(2592)))] / (2 * 1)
After simplifying, we get:
y = [144 ± √(20736 - 10368)] / 2
y = [144 ± √(10368)] / 2
y = [144 ± 102.384] / 2
y = 246.384 / 2 or y = 41.616 / 2
y = 123.192 or y = 20.808
Since lengths cannot be negative, we discard the negative solution, y = 20.808.
Finally, to find the area of the triangle, we use the formula for the area of a right triangle:
Area = (1/2) * base * height
Since the triangle is an isosceles right triangle, the two legs are equal. So the area can be expressed as:
Area = (1/2) * x * x
Substituting the value of x when y = 123.192 into the equation:
Area = (1/2) * (36 - 123.192) / 2 * (36 - 123.192) / 2
Area = (1/2) * 13.904 * 13.904
Area ≈ 95.991 square meters
Therefore, the area of the triangle solely in terms of the hypotenuse y, when y = 123.192, is approximately 95.991 square meters.
y^2 = x^2 + x^2 = 2x^2
area = 1/2 x*x = 1/2 x^2 = 1/2(y^2/2) = y^2/4