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?

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.

Step-by-step 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.