Broken tree

Geometry Level 1

During the hurricane a 16-meter-tall tree got fractured and its higher part fell down so that its tip touched the ground at the distance of 8 meters from the trunk. At what height is the fracture?


The answer is 6.

Maria Paszkiewicz by Maria Paszkiewicz , 29, Germany

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2 solutions

Maria Paszkiewicz
Apr 14, 2020

I solved the task using ruler-and-compass construction. I am going to show the steps below: 1. This is the initial problem: 2. Draw circles of the same radius and the centers of points A and C, so that the circles intersect: 3. Connect the points D and E. The segment DE is a bisector of the line segment AC 4. The searched height is the segment BF, wchich is 6 units long: It is so because the segments AF and CF have equal length, which sum up to the initial height of the tree.

2 Helpful 6 Interesting 0 Brilliant 0 Confused

I would urge caution when combining a ruler-and-compass construction with a distance estimation.

Richard Desper - 1 year, 1 month ago

The task comes from a text book for mathematics in fifth grade (I only changed the numbers to get an integer as a solution). It appears that the pupils learn ruler-and-compass construction first, before they get to know Pythagorean theorem. This is actually the reason why I have made the construction to find a solution. I think it is an interesting approach too, so I wanted to show it.

Maria Paszkiewicz - 1 year, 1 month ago
Chris Lewis
Apr 14, 2020

Say the fracture is at a height h h from the ground. The length of the broken top part of the tree is 16 − h 16-h . By Pythagoras, h 2 + 8 2 = ( 16 − h ) 2 h^2+8^2=(16-h)^2

Expanding, h 2 + 64 = 256 − 32 h + h 2 h^2+64=256-32h+h^2

Cancelling and rearranging, 32 h = 192 32h=192

Solving, h = 6 h=\boxed6

6 Helpful 0 Interesting 0 Brilliant 0 Confused

Did it the same way.

Richard Desper - 1 year, 1 month ago

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