A 13 feet ladder is leaning against a house. The distance from the bottom of the ladder to the house is 7 feet less than the distance from the top of the ladder to the ground. Find the distance between the bottom of the ladder and the house.
Let's assume the distance from the bottom of the ladder to the house is 'x' feet.
According to the problem, the distance from the top of the ladder to the ground is x + 7 feet.
Using the Pythagorean theorem, we can find the length of the ladder:
ladder^2 = (x + 7)^2 + x^2
Simplifying this equation, we get:
169 = 2x^2 + 14x + 49
Rearranging the equation to 2x^2 + 14x + 49 - 169 = 0:
2x^2 + 14x - 120 = 0
Factoring the quadratic equation, we get:
2(x^2 + 7x - 60) = 0
2(x + 12)(x - 5) = 0
Setting each factor to zero and solving for 'x', we find two possible solutions:
x + 12 = 0 or x - 5 = 0
x = -12 or x = 5
Since we cannot have a negative distance, the distance from the bottom of the ladder to the house is 5 feet.
To find the distance between the bottom of the ladder and the house, we can use the Pythagorean theorem. Let's assume the distance from the bottom of the ladder to the house is x.
According to the given information, the distance from the top of the ladder to the ground is x + 7 (since it is 7 feet more than the distance to the house).
Using the Pythagorean theorem, we have:
(13 feet)^2 = (x + 7 feet)^2 + x^2
Simplifying the equation:
169 = x^2 + 14x + 49 + x^2
169 = 2x^2 + 14x + 49
Rearranging the equation to form a quadratic equation:
2x^2 + 14x + 49 - 169 = 0
2x^2 + 14x - 120 = 0
Now, we can solve this quadratic equation. Factoring out a common factor of 2:
2(x^2 + 7x - 60) = 0
Factoring further, we have:
2(x - 5)(x + 12) = 0
Setting each factor equal to zero, we have two possible solutions:
x - 5 = 0 or x + 12 = 0
Solving for x in each case:
x = 5 or x = -12
Since distance cannot be negative, we discard the second solution.
Therefore, the distance between the bottom of the ladder and the house is x = 5 feet.
x^2 + (x+7)^2 = 13^2
Looks like a 5-12-13 triangle to me.