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 and 50ft from the ground. The second ladder is 25ft from the ground.
To determine the length of the short ladder, we need to assume the wall and the ground form a right angle, and both ladders make the same angle with the ground, forming two similar right triangles.
Let's call the lengths of the ladders L1 and L2 (L1 being the 80 ft ladder). Let's call the distances they reach up the wall H1 and H2 (H1 being the 50 ft height, and H2 being the 25 ft height). Lastly, let's call the distances from the base of the wall to the foot of each ladder D1 and D2.
For the 80 ft ladder (L1) reaching 50 ft up the wall (H1), we have:
L1 = 80 ft,
H1 = 50 ft.
The distance from the wall to the bottom of the ladder (D1) can be found using the Pythagorean Theorem:
L1^2 = H1^2 + D1^2.
Let's solve for D1:
D1^2 = L1^2 - H1^2,
D1^2 = 80^2 - 50^2,
D1^2 = 6400 - 2500,
D1^2 = 3900,
D1 = √3900,
D1 = 62.5 ft.
Now, for the shorter ladder, we know the height it reaches (H2):
H2 = 25 ft.
Because the ladders lean at the same angle, they form similar triangles, meaning the ratios of corresponding sides are equal:
H1/D1 = H2/D2.
We want to find the length of the shorter ladder (L2), so we substitute in the values we have, including D2, which we need to find first:
50/62.5 = 25/D2.
Now, solve for D2:
D2 = (25 * 62.5) / 50,
D2 = 1562.5 / 50,
D2 = 31.25 ft.
Now we have the distance D2 for the shorter ladder. We use the Pythagorean Theorem again to find L2:
L2^2 = H2^2 + D2^2,
L2^2 = 25^2 + 31.25^2,
L2^2 = 625 + 976.5625,
L2^2 = 1601.5625,
L2 = √1601.5625,
L2 ≈ 40 ft.
So, the short ladder is approximately 40 feet long.