Three colinear points on the coordinate plane are R(x,y) S(x+8h, y+8k) and P(x+6h,y+6k0
Determine value of RP/SP and RP/RS?
To determine the values of RP/SP and RP/RS, we need to find the distances between the given points.
The distance between two points (x₁, y₁) and (x₂, y₂) on the coordinate plane can be calculated using the distance formula:
d = √((x₂ - x₁)² + (y₂ - y₁)²)
Let's start by finding the distance between points R(x, y) and P(x + 6h, y + 6k).
Distance RP:
RP = √((x + 6h - x)² + (y + 6k - y)²)
RP = √(6h)² + (6k)²
RP = √(36h² + 36k²)
RP = 6√(h² + k²)
Next, let's find the distance between points R(x, y) and S(x + 8h, y + 8k).
Distance RS:
RS = √((x + 8h - x)² + (y + 8k - y)²)
RS = √(8h)² + (8k)²
RS = √(64h² + 64k²)
RS = 8√(h² + k²)
Finally, let's find the distance between points R(x, y) and P(x + 6h, y + 6k).
Distance SP:
SP = √((x + 6h - x + 8h)² + (y + 6k - y + 8k)²)
SP = √(14h)² + (14k)²
SP = √(196h² + 196k²)
SP = 14√(h² + k²)
Now, we can calculate the values of RP/SP and RP/RS:
RP/SP = (6√(h² + k²)) / (14√(h² + k²))
Simplifying by canceling out the common term in the numerator and denominator, we get:
RP/SP = 3/7
RP/RS = (6√(h² + k²)) / (8√(h² + k²))
Simplifying by canceling out the common term, we get:
RP/RS = 3/4
Therefore, RP/SP = 3/7 and RP/RS = 3/4 for the given points on the coordinate plane.