Answer:
a. (-3, 6)
b. 268.3 m
Step-by-step explanation:
a. Determine the coordinates of the tennis court.
Let (x,y) be the coordinates of the tennis court. Since we have equal distances from the tennis court to each residence, we have using the equation for distance between two points (x₁,y₁) and (x₂,y₂), d = √[(y₂ - y₁)² + (x₂ - x₁)²]. Using the first two residences, (-6, 12) and (0,0) as (x₂,y₂) and (x, y) = (x₁,y₁), So,
d = √[(12 - y)² + (-6 - x)²]
d = √[(12 - y)² + (6 + x)²]
d = √[(144 - 24y + y² + 36 + 12x + x²]
d = √x² + y² + 12x - 24y + 180
d² = x² + y² + 12x - 24y + 180 (1)
Also, with (x₂, y₂) = (0,0) we have
d = √[(0 - y)² + (0 - x)²]
d = √[( - y)² + (- x)²]
d = √[y² + x²]
d = √[x² + y²]
d² = x² + y² (2)
Also, with (x₂, y₂) = (3,9) we have
d = √[(9 - y)² + (3 - x)²]
d = √[(81 - 18y + y² + 9 - 6x + x²]
d = √(x² + y² - 6x - 18y + 90)
d² = x² + y² - 6x - 18y + 90 (3)
Substituting (2) into (1), we have
d² = x² + y² + 12x - 24y + 180 (1)
d² = d² + 12x - 24y + 180 (1)
d² - d² = 12x - 24y + 180
12x - 24y + 180 = 0
12x - 24y = -180
dividing through by 12, we have
x - 2y = -15 (4)
Substituting (2) into (3), we have
d² = x² + y² - 6x - 18y + 90 (3)
d² = d² - 6x - 18y + 90 (3)
d² - d² = - 6x - 18y + 90
- 6x - 18y + 90 = 0
- 6x - 18y = -90
dividing through by -6, we have
x + 3y = 15 (5)
Subtracting (5) from (4), we have
x - 2y = -15 (4)
-
x + 3y = 15 (5)
-5y = -30
y = -30/-5
y = 6
substituting y = 6 into (4), we have
x - 2y = -15
x - 2(6) = -15
x - 12 = -15
x = 12 - 15
x = -3
So, the coordinates of the tennis court is (-3, 6)
b. What is the distance from a residence to the court to the nearest 10th of a metre? (1unit=40m).
Using equation (2),
d² = x² + y² (2) where x and y are the coordinates of the tennis court, and x = -3 and y = 6
So,
d² = x² + y²
d² = (-3)² + (6)²
d² = 9 + 36
d² = 45
d = √45 units
Since 1 unit = 40 m,
d = √45 × 40 m
d = 6.708 × 40 m
d = 268.33 m
d ≅ 268.3 m to the nearest tenth of a metre
