The points (-10,5) and (4,5) are two vertices of a rectangle. Find two other points that make a rectangle with a perimeter of 44 units.
make a sketch, notice the 2 given points form a horizontal line, and has length 14
So the other side must be a vertical line
let the point below (4,5) be (4,y)
then (5-y) + 14 = 22
y = -3
So the point below (4,5) is (4,-3)
and the point below (-10,5) is (-10, -3)
the point could also be above (4,5)
then y-5 + 14 = 22
y = 13
and the two points are (4,13) and (-10,13)
To find two other points that make a rectangle with a perimeter of 44 units, we can use the fact that opposite sides of a rectangle are equal in length.
The distance between the two given points (-10,5) and (4,5) represents the length of one side of the rectangle, which is 4 - (-10) = 14 units.
Let's denote the other two points' coordinates as (x1, y1) and (x2, y2).
Since opposite sides of a rectangle are equal in length, the distance between (x1, y1) and (x2, y2) should also be 14 units.
However, we also need to consider the perimeter of the rectangle, which is the sum of the four side lengths.
The given perimeter is 44 units, so the sum of the four side lengths is 44 units. If one side length is 14 units, the sum of the other three side lengths must be 44 - 14 = 30 units.
To find two other points that satisfy these conditions, we can experiment with different combinations of side lengths. Let's start with two sides of length 14 units and two sides of length 1 unit.
We can choose the following coordinates for the two other points:
(x1, y1) = (-10, 5 + 1) = (-10, 6)
(x2, y2) = (4 + 1, 5) = (5, 5)
By connecting these four points (-10,5), (4,5), (-10,6), and (5,5), we can visualize the rectangle.
To confirm that these points form a rectangle with a perimeter of 44 units, we need to calculate the distances between them.
Distance between (-10,5) and (4,5): √[(4 - (-10))^2 + (5 - 5)^2] = √[196 + 0] = √196 = 14 units
Distance between (-10,5) and (-10,6): √[(-10 - (-10))^2 + (6 - 5)^2] = √[0 + 1] = √1 = 1 unit
Distance between (-10,6) and (5,5): √[(5 - (-10))^2 + (5 - 6)^2] = √[225 + 1] = √226 ≈ 15 units
Distance between (5,5) and (4,5): √[(4 - 5)^2 + (5 - 5)^2] = √[1^2 + 0] = √1 = 1 unit
The sum of the distances between the points is 14 + 1 + 15 + 1 = 31 units, which does not match the given perimeter of 44 units.
Hence, this combination of side lengths and points is not correct.
To find the correct combination, we can try different side lengths and combinations of coordinates until we find one that satisfies both the condition of equal side lengths and a total perimeter of 44 units.