Two ladders are leaning against a wall at the same angle, as shown. How long is the short ladder?
the first ladder is 80ft long. The second ladder is 50ft long and 25ft from the ground.
30 ft
50 ft
40 ft
60 ft
To find the length of the short ladder, we can use the concept of similar triangles.
Let's call the height of the wall "h" and the distance between the base of the ladder and the wall "d".
In the given scenario, we have two similar triangles: one formed by the first ladder and the wall, and the other formed by the second ladder and the wall.
In the first triangle:
Length of the ladder = 80 ft
Height of the wall = h
Distance between the base of the ladder and the wall = d
In the second triangle:
Length of the ladder = 50 ft
Height of the wall = 25 ft
Distance between the base of the ladder and the wall = d
Using the concept of similar triangles, we can write the following proportion:
(Length of the first ladder) / (Height of the wall) = (Length of the second ladder) / (Height of the wall)
Using the given values, we can plug them into the proportion and solve for the unknown length of the short ladder:
80 ft / h = 50 ft / 25 ft
Cross-multiplying, we get:
(80 ft)(25 ft) = (50 ft)(h)
2000 ft^2 = 50 ft * h
Dividing both sides by 50 ft, we get:
h = 2000 ft^2 / 50 ft
h = 40 ft
Therefore, the height of the wall is 40 ft. Since the two ladders are leaning against the wall at the same angle, the distance between the base of the ladder and the wall is also the same for both ladders, which is 40 ft.
Therefore, the short ladder is 40 ft long.