Answer:
The coordinates for the location of the center of the platform are (-1 , 2)
Step-by-step explanation:
* Lets revise the equation of the circle
- The equation of the circle of center (h , k) and radius r is:
(x - h)² + (y - k)² = r²
- The center is equidistant from any point lies on the circumference
of the circle
- There are three points equidistant from the center of the circle
- We have three unknowns in the equation of the circle h , k , r
- We will substitute the coordinates of these point in the equation of
the circle to find h , k , r
* Lets solve the problem
∵ The equation of the circle is (x - h)² + (y - k)² = r²
∵ Points D (-2 , -4) , E (1 , 5) , F (2 , 0)
- Substitute the values of x and y b the coordinates of these points
# Point D (-2 , -4)
∵ (-2 - h)² + (-4 - k)² = r² ⇒ (1)
# Point E (1 , 5)
∵ (1 - h)² + (5 - k)² = r² ⇒ (2)
# Point (2 , 0)
∵ (2 - h)² + (0 - k)² = r²
∴ (2 - h)² + k² = r² ⇒ (3)
- To find h , k equate equation (1) , (2) and equation (2) , (3) because
all of them equal r²
∵ (-2 - h)² + (-4 - k)² = (1 - h)² + (5 - k)² ⇒ (4)
∵ (1 - h)² + (5 - k)² = (2 - h)² + k² ⇒ (5)
- Simplify (4) and (5) by solve the brackets power 2
# (a ± b)² = (a)² ± (2 × a × b) + (b)²
# Equation (4)
∴ [(-2)² - (2 × 2 × h) + (-h)²] + [(-4)² - (2 × 4 × k) + (-k)²] =
[(1)² - (2 × 1 × h) + (-h)²] + [(5)² - (2 × 5 × k) + (-k)²]
∴ 4 - 4h + h² + 16 - 8k + k² = 1 - 2h + h² + 25 - 10k + k² ⇒ add like terms
∴ 20 - 4h - 8k + h² + k² = 26 - 2h - 10k + h² + k² ⇒ subtract h² and k²
from both sides
∴ 20 - 4h - 8k = 26 - 2h - 10k ⇒ subtract 20 and add 2h , 10k
for both sides
∴ -2h + 2k = 6 ⇒ (6)
- Do the same with equation (5)
# Equation (5)
∴ [(1)² - (2 × 1 × h) + (-h)²] + [(5)² - (2 × 5 × k) + (-k)²] =
[(2)² - (2 × 2 × h) + k²
∴ 1 - 2h + h² + 25 - 10k + k² = 4 - 4h + k²⇒ add like terms
∴ 26 - 2h - 10k + h² + k² = 4 - 4h + k² ⇒ subtract h² and k²
from both sides
∴ 26 - 2h - 10k = 4 - 4h ⇒ subtract 26 and add 4h
for both sides
∴ 2h - 10k = -22 ⇒ (7)
- Add (6) and (7) to eliminate h and find k
∴ - 8k = -16 ⇒ divide both sides by -8
∴ k = 2
- Substitute this value of k in (6) or (7)
∴ 2h - 10(2) = -22
∴ 2h - 20 = -22 ⇒ add 20 to both sides
∴ 2h = -2 ⇒ divide both sides by 2
∴ h = -1
* The coordinates for the location of the center of the platform are (-1 , 2)
