A right triangle has perimeter equal to 80 and hypotenuse equal to 34. What is the area of the triangle?

let one leg be x

then the other leg is 80 - 34 - x = 46-x

x^2 + (46-x)^2 = 34^2
x^2 + 2116 - 92x + x^2 = 1156
2x^2 - 92x + 960 = 0
x^2 - 46x + 480 = 0
(x -30)(x-16) = 0
x = 30 or x = 16
if x=30, other leg is 46-30 = 16
if x = 16 , then other leg is 46-16 = 30
(symmetrical solution)

area of triangle = (1/2)(16)(30) = 240 square units

Thanks! :)

I thought it made sense to make it into a quadratic equation.

Stupid me... I didn't factorise correctly.

To find the area of a right triangle, we need to know the lengths of the two legs (the sides adjacent to the right angle).

Let's denote the lengths of the two legs as a and b, and the length of the hypotenuse as c.

In this case, we are given the perimeter (the sum of all three sides) of the right triangle. The perimeter is equal to the sum of the lengths of the two legs and the hypotenuse. So, we can set up the equation:

a + b + c = 80

We are also given the length of the hypotenuse, which is 34.

c = 34

Substituting this value into the equation, we get:

a + b + 34 = 80

Now, we can solve for a and b:

a + b = 80 - 34
a + b = 46

We have the equation a + b = 46, representing the sum of the lengths of the two legs.

Since this is a right triangle, we can use the Pythagorean theorem, which states that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the two legs (a and b):

a^2 + b^2 = c^2

Substituting the values we know:

a^2 + b^2 = 34^2
a^2 + b^2 = 1156

Now we have two equations, a + b = 46 and a^2 + b^2 = 1156. We can solve these equations simultaneously to find the values of a and b.

One way to solve this is to eliminate one variable by substitution. We can solve the first equation for a (a = 46 - b) and substitute it into the second equation:

(46 - b)^2 + b^2 = 1156
2116 - 92b + b^2 + b^2 = 1156
2b^2 - 92b + 2116 - 1156 = 0
2b^2 - 92b + 960 = 0

Now we can solve this quadratic equation for b using factoring, completing the square, or using the quadratic formula:

(b - 20)(2b - 48) = 0

From here, we have two possible values for b: b = 20 or b = 24.

Substituting these values back into the first equation, we can find the corresponding values for a:

When b = 20, a + 20 = 46, a = 26
When b = 24, a + 24 = 46, a = 22

Therefore, the lengths of the two legs are a = 26, b = 20, or a = 22, b = 24.

To find the area of the triangle, we use the formula:

Area = 0.5 * base * height

Since this is a right triangle, either a or b can be considered the base, and the other one is the height.

When a = 26 and b = 20, the area = 0.5 * 26 * 20 = 260 square units.

When a = 22 and b = 24, the area = 0.5 * 22 * 24 = 264 square units.

Therefore, the area of the triangle is either 260 square units or 264 square units, depending on the lengths of the legs.