Answer:
the coordinates of point C are approximately (0.56, 17.72)
Step by step Explanation:
To find the coordinates of point C, we need to find its distance from point A as a fraction of the distance between points A and B, and then use that fraction to find the coordinates of C.
Let's call the distance between points A and B "d". We can find the distance using the distance formula:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
where (x₁, y₁) and (x₂, y₂) are the coordinates of points A and B, respectively. Plugging in the values, we get:
d = √[(3 - (-5))² + (12 - 16)²] = √64 + 16 = √80
Now, let's call the fraction of the distance from A to B that C lies on "f". We know that C is closer to A than to B, so 0 ≤ f ≤ 1. We can find f using the formula:
f = AC/AB
where AC is the distance from A to C. We can use the distance formula again to find AC:
AC = √[(x - (-5))² + (y - 16)²]
where (x, y) are the coordinates of point C. We also know that C is on the line segment AB, so its x-coordinate must satisfy:
(x - (-5))/(3 - (-5)) = f
Simplifying, we get:
(x + 5)/8 = f
Solving for x, we get:
x = 8f - 5
Similarly, we can use the y-coordinate of C to form another equation:
(y - 16)/(12 - 16) = f
Simplifying and solving for y, we get:
y = 4(1 - f) + 16 = 20 - 4f
Now we have expressions for x and y in terms of f. We can substitute these into the expression for AC:
AC = √[(8f - 5 - (-5))² + (20 - 4f - 16)²] = √[64f² - 64f + 100]
Finally, we can plug in our expressions for AC and AB to solve for f:
f = AC/AB = √[64f² - 64f + 100]/√80
Squaring both sides and simplifying, we get a quadratic equation:
64f² - 64f + 100 = 80f²
16f² + 64f - 100 = 0
Dividing both sides by 4 and using the quadratic formula, we get:
f = (-64 ± √(64² - 4(16)(-100))) / 2(16)
f ≈ -0.32 or f ≈ 1.57
Since 0 ≤ f ≤ 1, the valid solution is:
f ≈ 0.57
Now we can use this value to find the coordinates of C:
x = 8f - 5 ≈ 0.56
y = 20 - 4f ≈ 17.72
Therefore, the coordinates of point C are approximately (0.56, 17.72)