boomtangboy wrote:
A ladder 25 feet long is leaning against a wall that is perpendicular to level ground. The bottom of the ladder is 7 feet from the base of the wall. If the top of the ladder slips down 4 feet, how many feet will the bottom of the ladder slip?
(A) 4
(B) 5
(C) 8
(D) 9
(E) 15
A ladder 25 feet long is leaning against a wall that is perpendicular to level ground. The bottom of the ladder is 7 feet from the base of the wall. We have something like this:
Since the wall is perpendicular to the ground, we have a right triangle, which means we can apply the Pythagorean Theorem to write: x² + 7² = 25²
Simplify to get: x² + 49 = 625
Subtract 49 from both sides: x² = 576
Solve:
x = 24ASIDE: We could have avoided all of the calculations above had we recognized that 7 and 25 are two of the three values in the Pythagorean triple
7-24-25, which means the missing side must have length
24So, the ladder, in its ORIGINAL position, extends to a height of
24 feet.
If the top of the ladder slips down 4 feet . . . If the ladder slips down 4 feet, then the ladder's NEW height =
24 - 4 =
20So, we have something like this:
Once again, we COULD apply the Pythagorean Theorem to find the value of y (I'll let you do that on your own)
However, we can save time by recognizing that 25 and 20 are two of the three values in the
magnified version of the Pythagorean triple
3-4-5That is, if we take the Pythagorean triple 3-4-5 and multiply each side length by 5, we get the equivalent Pythagorean triple
15-20-25This means the missing side must have length 15
In other words,
y = 15. . . how many feet will the bottom of the ladder slip?Originally, the bottom of the ladder was
7 feet from the wall.
Afterwards, the bottom of the ladder was
15 feet from the wall
So, the bottom slipped 8 feet
Answer: C