A right triangle has an area of 84 ftand a hypotenuse 25 ft long. What are the lengths of its two sides?
Ive gotten this far:
25^2= a^2+b^2
a=25-b
84= ab/2
168= 25b-b^2
b^2-25b+168=0
I then use the quad. formlua to solve for b... but I get an error. Can someone help me?
25^2= a^2+b^2... so far so good.
"a=25-b" ... problem stars here!
Start with
a²+b²=25²
and
(1/2)ab=84
2ab=4*84=336
So
a²+b²+2ab = 625+336 = 961
(a+b)²=961
a+b=√961=31
ab=2*84=168
a(31-a)=168
a²-31a+168=0
Solve quadratic in a to get
a=24, or a=7
So the two sides are 24 and 7
To find the lengths of the two sides of the right triangle, we can use the Pythagorean theorem and the formula for the area of a triangle.
Let's start by using the formula for the area of a triangle:
Area = (1/2) * base * height
In this case, the area is given as 84 ft^2. Let's assume one of the sides of the right triangle is the base, and the other side is the height.
Area = (1/2) * base * height
84 = (1/2) * base * height
168 = base * height
Now, let's use the Pythagorean theorem to relate the sides of the right triangle:
a^2 + b^2 = c^2
Here, a and b represent the lengths of the two sides of the right triangle, and c represents the length of the hypotenuse, which is given as 25 ft.
a^2 + b^2 = 25^2
a^2 + b^2 = 625
Now we have two equations:
168 = base * height
a^2 + b^2 = 625
Since the problem states that the hypotenuse is 25 ft long, we can eliminate the need to solve for b and use substitution to solve for a.
We can rearrange the equation a^2 + b^2 = 625 to express a^2 in terms of b:
a^2 = 625 - b^2
Now, we substitute this expression for a^2 into the equation 168 = base * height:
168 = b * (625 - b^2)
Expanding the equation gives:
168 = 625b - b^3
Rearranging the equation gives:
b^3 - 625b + 168 = 0
Now, we can use a numerical method such as trial and error, or use a graphing calculator or computer software to find the solutions for b.
To find the lengths of the two sides of the right triangle, you can use the Pythagorean theorem and the formula for the area of a triangle. Here's how you can solve it step by step:
1. Let's assume the two sides of the right triangle are represented by 'a' and 'b', where 'a' is the shorter side and 'b' is the longer side.
Now, we know that the area of a right triangle is A = (ab)/2, where A is the area, 'a' and 'b' are the lengths of its two sides.
2. From the given information, the area of the right triangle is 84 ft². So, we have the equation (ab)/2 = 84.
3. We are also given that the hypotenuse (the longest side) of the right triangle is 25 ft. According to the Pythagorean theorem, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. Applying this, we get the equation a² + b² = 25².
Now, let's solve these equations simultaneously:
Step 1: Substitute a = 25 - b into the area equation.
(25 - b) * b / 2 = 84
Step 2: Simplify the equation.
25b - b² = 168
Step 3: Rearrange the equation to solve for b.
b² - 25b + 168 = 0
Now, to solve this quadratic equation for 'b', you can either factorize it or use the quadratic formula.
Factoring:
The equation b² - 25b + 168 = 0 can be factored as (b - 8)(b - 21) = 0.
So, either b - 8 = 0 or b - 21 = 0.
If b - 8 = 0, then b = 8.
If b - 21 = 0, then b = 21.
Therefore, the two possible values for 'b' are 8 and 21.
Now, substitute these values of 'b' back into the equation a = 25 - b to find the corresponding values of 'a'.
If b = 8:
a = 25 - 8
a = 17
If b = 21:
a = 25 - 21
a = 4
So, the two possible lengths of the sides of the right triangle are:
a = 17 ft, b = 8 ft
or
a = 4 ft, b = 21 ft.
Therefore, there are two valid triangles that satisfy the given conditions.