A pole is 8 feet long. Its shadow is 6 feet long. If you draw a line from the tip of the pole to the tip of the shadow, how long will the line be?
To find the length of the line from the tip of the pole to the tip of the shadow, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In this case, the pole acts as the height of the right triangle, and the shadow acts as the base. The line from the tip of the pole to the tip of the shadow acts as the hypotenuse.
Let's solve for the length of the line:
Using the Pythagorean theorem:
hypotenuse^2 = height^2 + base^2
Line^2 = 8^2 + 6^2
Line^2 = 64 + 36
Line^2 = 100
Taking the square root of both sides:
√(Line^2) = √100
Line = 10 feet
Therefore, the line from the tip of the pole to the tip of the shadow is 10 feet long.
To find the length of the line from the tip of the pole to the tip of the shadow, we can use the concept of similar triangles. Similar triangles are triangles that have the same shape but possibly different sizes.
In this case, we have two similar triangles: the triangle formed by the pole, its shadow, and the line from the tip of the pole to the tip of the shadow, and the triangle formed by the pole, its height, and the line from the tip of the pole to the tip of the shadow.
The two triangles are similar because they have a pair of congruent angles. The angle formed by the pole and its shadow is the same as the angle formed by the pole and its height.
Using the concept of proportional sides in similar triangles, we can set up a proportion to find the length of the line from the tip of the pole to the tip of the shadow.
Let x be the length of the line from the tip of the pole to the tip of the shadow.
Then we have the proportion:
x / 6 = 8 / h
where h is the height of the pole.
We know that the pole is 8 feet long, so we can substitute this value:
x / 6 = 8 / 8
Simplifying the fraction, we have:
x / 6 = 1
To solve for x, we can cross multiply:
x = 6
Therefore, the length of the line from the tip of the pole to the tip of the shadow is 6 feet.
This is a direct application of Pythagoras
length of line ---- h
solve for h
h^ = 6^2 + 8^2