Garden Perimeter

A garden has the shape of a right-angled triangle with sides of 30, 40, 50. A straight fence goes from the corner with the right-angle to a point on the opposite side, dividing the garden into two sections which have the same perimeter.

How long is the fence?

[This question appears on a 13+ exam paper and I cannot work out the solution for the pupils without using trigonometry (which they have not yet studied). Any help would be greatly appreciated.]

#Geometry

Easy Math Editor

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Comments

We want to split the right triangle into two sections with equal perimeter. Obviously, the fence will cut the hypotenuse into two line segments of length 20 and 30 in order for both the sections to have an equal perimeter. The problem is that we need to find the length of the fence without using trigonometry. We know that the fence is not the altitude unless the length of the fence is 24. This is because 0.5(30)(40) = 0.5(50)(length of fence). If the length of the fence is not 24, we can construct the altitude to the hypotenuse knowing that it will not be the same line as the fence. The endpoint of the altitude on the hypotenuse will be closer to the larger angle of the triangle than the smaller angle. That means the line segment will cut into the 30 part of the hypotenuse of the big triangle. The altitude will cut the big right triangle into 2 smaller right triangles, one of dimensions 24 (the length of the altitude), 40 (the length of the side of the bigger triangle but the hypotenuse of the smaller triangle), and 20 + x (since the altitude cuts into the 30 section of the big triangle, the second side of the smaller triangle is 20 plus an additional amount). We can solve for x. If the triangle does not exist, then the fence is clearly the altitude. If it does exist, we realize that the altitude also creates a right triangle with a portion of the hypotenuse (of length x), the altitude, and the fence. Finally, we can use Pythagorean's theorem to solve for the length of the fence (the hypotenuse of the smallest right triangle).

A Former Brilliant Member - 3 years, 3 months ago

Thank you for your guidance.

Ben Bazzard - 3 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$