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$25$ . A tree $32$ meter-high is broken off $7$ meter from the ground. How far from the foot of the tree will the top strike the ground?

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Pythagorean theorem

Full solution

Q.

25

. A tree

32

meter-high is broken off

7

meter from the ground. How far from the foot of the tree will the top strike the ground?

Visualize Triangle Problem: Visualize the problem as a right triangle where the tree is the hypotenuse. The tree breaks $7$ meters from the ground, creating a right triangle with the ground and the remaining part of the tree.
Determine Triangle Sides: Determine the lengths of the sides of the triangle. $\newline$ The full height of the tree is $32$ meters, and it breaks $7$ meters from the ground, so the length of the tree from the break to the top is $32 - 7 = 25$ meters.
Use Pythagorean Theorem: Use the Pythagorean Theorem to find the distance from the foot of the tree to where the top strikes the ground. $\newline$ Let's denote the distance from the foot of the tree to where the top strikes the ground as $d$ . The broken part of the tree ( $25$ meters) is the hypotenuse, and the height from the ground to the break point ( $7$ meters) is one leg of the right triangle. We need to find the other leg $d$ .
Apply Theorem: Apply the Pythagorean Theorem. $\newline$ The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse $c$ is equal to the sum of the squares of the lengths of the other two sides $a$ and $b$ . $\newline$ So, we have: $\newline$ $a^2 + b^2 = c^2$ $\newline$ $d^2 + 7^2 = 25^2$
Solve for Distance: Plug in the known values and solve for $d$ . $\newline$ $d^2 + 49 = 625$ $\newline$ $d^2 = 625 - 49$ $\newline$ $d^2 = 576$
Take Square Root: Take the square root of both sides to solve for ＂d＂. $\newline$ $\sqrt{d^2} = \sqrt{576}$ $\newline$ $d = 24$

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