With a 12ft. pole marked in feet, how can one determine the foot of the perpendicular let fall to the floor from a point on a ceiling of a room 9ft. high?
To determine the foot of the perpendicular that falls to the floor from a point on the ceiling of a room using a 12ft pole, you can use similar triangles and the concept of proportional ratios. Here's how:
1. Start by visualizing the situation: You have a room with a 9ft high ceiling and a 12ft pole. Let's call the point on the ceiling where the perpendicular falls "Point A" and let's call the foot of the perpendicular on the floor "Point B".
2. Now, create a triangle: Draw a vertical line segment of length 9ft from Point A, representing the ceiling height. Then, draw a horizontal line segment starting from Point A, with a length that represents the distance from the pole to the point directly above Point B. Let's call this distance "x".
3. Recognize the similarity: We have two triangles in this scenario – the larger triangle formed by the 9ft ceiling height and the smaller triangle formed by the distance x and the foot of the perpendicular. These two triangles are similar because they have a pair of congruent angles (the right angles) and a pair of proportional sides.
4. Set up the proportion: Since the two triangles are similar, we can set up a proportion between their corresponding sides. The side lengths we are interested in are 9ft (ceiling height) and x (distance from pole to Point B).
Proportion: 9ft / 12ft = x / 12ft
Explanation: The proportion states that the ratio of the corresponding sides in the larger triangle (9ft/12ft) is equal to the ratio of the corresponding sides in the smaller triangle (x/12ft).
5. Solve for x: To find x, we can cross-multiply the proportion and solve for x. Cross-multiplying means multiplying the numerator of the left side with the denominator of the right side, and vice versa.
(9ft * 12ft) = (12ft * x)
Now, simplify the equation:
108ft = 12ft * x
Divide both sides of the equation by 12ft:
x = 9ft
6. Interpret the result: The value of x is 9ft, which means the distance from the pole to the point directly above Point B is 9ft. Therefore, the perpendicular falls to the floor 9ft away from the pole.
So, the foot of the perpendicular will be located 9ft away from the pole along the floor.