a ladder is resting against a wall. the top of the ladder touches the wall at a height of 6 feet. find the length of the ladder if the lenght is 2 feet more than its distance from the wall.
A rt. triangle is formed:
X Ft. = hor. side.
6 Ft. = ver. side.
(x+2) Ft. = hyp.
x^2 + 6^2 = (x+2)^2.
x^2 + 36 = x^2 + 4x + 4.
x^2 - x^2 + 4x + 4 = 36.
4x + 4 = 36.
4x = 36 - 4 = 32.
X = 32 / 4 = 8. Ft.
X+2 = 8 + 2 = 10 Ft .
To find the length of the ladder, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Let's denote the distance from the wall to the base of the ladder as "x". According to the problem, the length of the ladder is 2 feet more than this distance, so the length can be represented as "x + 2".
We can now write the equation for the Pythagorean theorem as:
(x + 2)^2 = x^2 + 6^2
Expanding the left side of the equation:
x^2 + 4x + 4 = x^2 + 36
Simplifying the equation by canceling out the common term "x^2" from both sides:
4x + 4 = 36
Next, subtracting 4 from both sides of the equation:
4x = 32
Dividing both sides of the equation by 4:
x = 8
Therefore, the distance from the wall to the base of the ladder is 8 feet. Since the length of the ladder is 2 feet more than this distance, the length of the ladder must be 8 + 2 = 10 feet.