Garden fence
Weekly Problem 44 - 2009
A garden has the shape of a right-angled triangle. A fence goes from the corner with the right-angle to a point on the opposite side. How long is the fence?
A garden has the shape of a right-angled triangle. A fence goes from the corner with the right-angle to a point on the opposite side. How long is the fence?
Age
14 to 16
Challenge level
Problem
A garden has the shape of a right-angled triangle with sides of length $30$, $40$ and $50$. 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?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
In the diagram below, triangle $ABC$ represents the garden, $CD$ represents the fence and $E$ is the foot of the perpendicular from $D$ to $AC$.
Image
The two sections of the garden have the same perimeter so $AD$ is $10$m longer than $DB$. Hence $AD=30$m and $DB=20$m.
Triangles $AED$ and $ACB$ are similar so $\frac{AE}{AC}=\frac{AD}{AB}=\frac{30}{50}$. Hence $AE=\frac{3}{5}\times 30$m $=18$m. So $EC=(30-18)$m $=12$m.
Also $\frac{ED}{CB}=\frac{AD}{AB}=\frac{30}{50}$. Hence $ED=\frac{3}{5}\times 40$m $=24$m.
Finally, by Pythagoras: $CD^2=EC^2+ED^2=(12^2+24^2)$m$^2=5\times12^2$m$^2$. So the length of the fence is $12\sqrt{5}$m.