The area of a right triangle is 12 square inches, and the sum of the lengths of the two legs, a and b, is 16 inches. Write a system that could be used to find the lengths of the legs of the right triangle.
ab/2 = 12
a+b = 16
To find the lengths of the legs of the right triangle, we can use the Pythagorean theorem and the given information. Let's denote one leg as "a" and the other leg as "b".
1. Start by writing down the Pythagorean theorem:
a^2 + b^2 = c^2, where c is the length of the hypotenuse.
2. Since we know that the area of the triangle is 12 square inches, we can use the formula for the area of a right triangle:
Area = (a * b) / 2
Substituting the given area, we have 12 = (a * b) / 2.
3. The sum of the lengths of the two legs is given as 16 inches:
a + b = 16.
Now we have a system of two equations that can be used to solve for the lengths of the legs. Let's solve step-by-step:
From equation (3), we can rewrite it as a = 16 - b.
Substituting this value of a into equation (2), we have:
12 = (16 - b) * b / 2.
Simplifying this equation, we get:
24 = (16 - b) * b.
Expanding and rearranging the terms, we have:
24 = 16b - b^2.
Rearrange this equation in standard quadratic form:
b^2 - 16b + 24 = 0.
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula.
Once we find the value of b, we can substitute it back into equation (3) to get the value of a.
Note: There may be two possible solutions for a and b, as there can be two different right triangles with the same conditions.