Answer: To find the coordinates of P, we will use the part-to-whole ratio of 1 to 5. This means that the distance from A to P is 1/6 of the distance from A to B.
We can find the distance from A to B by using the distance formula:
d = √((x2-x1)² + (y2-y1)²)
where (x1, y1) and (x2, y2) are the coordinates of A and B, respectively.
So, the distance from A to B is:
d = √((1-(-9))² + (8-3)²) = √((10)² + (5)²) = √(100 + 25) = √125 = 5√5
Since P partitions AB with the ratio 1 to 5, the distance from A to P is 1/6 of the distance from A to B, which is (1/6) * (5√5) = √5
To find the coordinates of P, we will use this distance and the direction from A to B to find the coordinates of P.
The direction from A to B is:
(1 - (-9))/(8 - 3) = 10/5 = 2
So, the slope of line AB is 2. Therefore, the slope of line AP is -1/2 (perpendicular to AB)
We know that A = (-9, 3) and the slope of line AP = -1/2. We can use this to find the equation of the line AP:
y - 3 = -1/2(x + 9)
y = -1/2x - 3/2 + 3
y = -1/2x + 3/2
Since the point A is on the line, we can substitute the coordinates of A into the equation:
3 = -1/2*(-9) + 3/2
3 = 4.5
This is not true, so we can say that the equation is wrong.
To find the equation of line AP, we can use the point-slope form of a line:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope of the line.
So, the equation of line AP is:
y - 3 = -1/2(x + 9)
Now we know that the point P is on the line AP and the distance from A to P is √5.
We can use this information to find the coordinates of P by moving along the line AP by √5 units in the direction of P.
To move in the direction of P, we need to move parallel to the line, which means we need to keep the slope of the line same.
So we can move
Explanation: