Let's solve the problem step-by-step.
1. First, understand the given data:
- The tree is broken at a height of 5 meters from the ground.
- The top of the tree touches the ground 12 meters away from the base.
2. We can visualize this situation as forming a right triangle.
- The height at which the tree is broken (5 meters) forms one leg of the triangle.
- The distance from the base where the top touches the ground (12 meters) forms the other leg of the triangle.
- The hypotenuse of the triangle would be the part of the tree above the breaking point that is now lying on the ground.
3. To find the hypotenuse (the length of the part of the tree that is lying on the ground), we use the Pythagorean theorem:
- According to the Pythagorean theorem: [tex]\( a^2 + b^2 = c^2 \)[/tex]
- Where [tex]\( a \)[/tex] is one leg of the triangle (5 meters), [tex]\( b \)[/tex] is the other leg (12 meters), and [tex]\( c \)[/tex] is the hypotenuse.
[tex]\[
c^2 = 5^2 + 12^2
\][/tex]
[tex]\[
c^2 = 25 + 144
\][/tex]
[tex]\[
c^2 = 169
\][/tex]
[tex]\[
c = \sqrt{169}
\][/tex]
[tex]\[
c = 13
\][/tex]
Therefore, the part of the tree above the breaking point has a length of 13 meters.
4. To find the original height of the tree, we need to add the height of the broken part (5 meters) to the length of the part that is now lying on the ground (13 meters).
[tex]\[
\text{Original height of the tree} = 5 \text{ meters} + 13 \text{ meters}
\][/tex]
[tex]\[
\text{Original height of the tree} = 18 \text{ meters}
\][/tex]
Thus, the original height of the tree was 18 meters.
