Question

3. Mr. Matho has a garden in the shape of an equilateral triangle, He has planted a tree in each corner

of his garden; an oak tree, a maple tree, and a pine tree. In order to water his garden, he decides

that the best place to put a sprinkler would be in the middle of the garden, so it would be equidistant

from all three trees. If the perimeter of his garden is 36 m and the sprinkler is 2-3 m from the

fence joining the oak and maple tree, how far away is the sprinkler from any one of the trees?

Answer 1

The distance the sprinkler is away from any one tree is 6.32 m if it is 2 m from the fence and 6.71 m if it is 3 m from the fence.

Explanation:

Since the perimeter of the triangle P = a + b + c where a, b and c are the length of sides. Since the triangle is an equilateral triangle, a = b = c. So,

P = 3a

a = P/3

Since the perimeter = 36 m,

a = 36 m/3 = 12 m.

Since the sprinkler is 2 m from the fence joining the oak tree and maple tree, thee line perpendicular from the sprinkler to the fence is thus 2 m. This line cuts the side at the midpoint of the fence which is 12 m/2 = 6 m.

Since this is now a right-angled triangle, we now use Pythagoras' theorem to find the distance from the sprinkler to each tree which is

L = √[(2 m)² + (6 m)²]

= √[ 4 m² + 36 m²}

= √40 m²

= 2√10 m

= 6.32 m

If the fence is 3 m from the sprinkler, the distance

L = √[(3 m)² + (6 m)²]

= √[ 9 m² + 36 m²}

= √45 m²

= 3√5 m

= 6.71 m

So, the distance the sprinkler is away from any one tree is 6.32 m if it is 2 m from the fence and 6.71 m if it is 3 m from the fence.