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Problem of the Week

Problem C and Solution

Around the Farm


Rahul has a farm he wishes to fence. The farm is the pentagon \(ABCDE\) shown below. He knows that \(ABCD\) is a 140 m by 150 m rectangle. He also knows that \(E\) is 50 m from the side \(AB\) and 30 m from the side \(BC\).
Determine the length of \(AE\), the length of \(DE\), and the perimeter of pentagon \(ABCDE\).

Side B C has length 140 metres and side C D has length 150 metres. Point E is in the interior of rectangle A B C D. A line is drawn perpendicular to side B C through E indicating the distance from B C to E and similarly for the distance from A B to E.

NOTE: The Pythagorean Theorem states, “In a right-angled triangle, the square of the length of hypotenuse (the side opposite the right angle) equals the sum of the squares of the lengths of the other two sides”.
In the following right triangle, \(p^2=r^2+q^2\).

A right-angled triangle with hypotenuse of length p and remaining sides of length r and length q.


Let \(F\) be the point on \(AB\) with \(EF=50\) m.
Let \(H\) be the point on \(BC\) with \(EH=30\) m.
Extend \(EF\) to \(G\) on \(CD\).
Since \(ABCD\) is a rectangle and \(FG\) is perpendicular to \(AB\), then \(FG\) is perpendicular to \(CD\) and \(FGCB\) is a rectangle. Therefore, \(FB= EH=GC=30\) m.
Also, \(DG = AF = AB - FB = 150 - 30 = 120\) m.
Angle A F E and angle B H E are right angles and E G has length 90.

Since \(\triangle AFE\) and \(\triangle DGE\) are right-angled triangles, we can use the Pythagorean Theorem to determine the lengths of \(AE\) and \(DE\).

In \(\triangle AFE\), \[\begin{aligned} AE^2&=&AF^2+FE^2\\ &=&120^2+50^2\\ &=&14\,400+2500\\ &=&16\,900\\ AE&=&130\mbox{, since $AE > 0$}\\\end{aligned}\] Triangle A F E has a right angle at F. The length of side A F is 120 and the length of side E F is 50.

In \(\triangle DGE\), \[\begin{aligned} DE^2&=&DG^2+EG^2\\ &=&120^2+90^2\\ &=&14\,400+8100\\ &=&22\,500\\ DE&=&150\mbox{, since $DE > 0$}\end{aligned}\] Triangle E G D has a right angle at G. The length of side E G is 90 and the length of side D G is 120.

Therefore, \(AE=130\) m and \(DE=150\) m.
Also, the perimeter of pentagon \(ABCDE\) is equal to \(AB + BC + CD + DE + AE = 150 + 140 + 150 + 150 + 130= 720\) m.