Problem 9: Find a relationship between x and y so that x, y is equidistant from the two points -3, 4 and 0, Problem Find a relationship between x and y so that the triangle whose vertices are given by x, y , 1, 1 and 5, 1 is a right triangle with the hypotenuse defined by the points 1, 1 and 5, 1. Solution to Problem 3: Use the distance formula to write an equation in x.

Solution to Problem 4: Use midpoint formula. Solution to Problem 5: Use distance formula to find distance the D1 from 0, y to 4, -9 and the distance D2 from 0, y to 0, For the triangle to be a right triangle, D3 the largest of the three distances must be the length of the hypotenuse and all three sides must satisfy Pythagorean theorem.

Solution to Problem 7: Let D1 be the distance from -2, 1 to 1, 1 , D2 be the distance from -2, 1 to 1, 2 and D3 the distance from 1, 1 to 1, 2. We now use the distance formula to find these distances. We now use Pythagorean theorem to check that it is a right triangle with D2 the hypotenuse and D1 and D3 the sides of the triangle.

## Distance and midpoint calculator

Solution to Problem 8: We apply the distance formula. Determine the coordinates of the mid-point of this line segment. The coordinates of the mid-point could have been calculated directly as shown below.

AB The coordinates of A are 2, 7. The coordinates of B are 5, 3.

AC The coordinates of C are —5, 3. Find the coordinates of the mid-point of the line segment:. Determine the coordinates of the mid-point of the line segments below. When the interval is not parallel to one of the axes we take the average of the x -coordinate and the y -coordinate. This is proved below. Let M be the midpoint of the line AB.

## Midpoint Calculator - Symbolab

Hence the x-coordinate of M is the average of 1 and 5. The y coordinate of M is the average of 2 and 8. Thus the coordinates of the midpoint M are 3, 5. We can find a formula for the midpoint of any interval. Suppose that P x 1 , y 1 and Q x 2 , y 2 are two points and let M x , y be the midpoint. Hence the x -coordinate of M is the average of x 1 and x 2 , and y -coordinate of M is the average of y 1 and y 2.

The midpoint of an interval with endpoints P x 1 , y 1 and Q x 2 , y 2 is ,.

### Midpoint work with steps

Take the average of the x -coordinates and the average of the y -coordinates. Let the coordinates of B be x 1 , y 1. The gradient is a measure of the steepness of line. There are several ways to measure steepness. In coordinate geometry the standard way to define the gradient of an interval AB is where rise is the change in the y -values as you move from A to B and run is the change in the x -values as you move from A to B. We will usually the pronumeral m for gradient.

Given the points A 2, 1 and B 5, 6 :. Notice that as you move from A to B along the interval the y -value increases as the x -value increases.

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The gradient is positive. Given the points A 2, 7 and B 6, 1. Notice that in this case as we move from A to B the y value decreases as the x value increases. The gradient is negative. If the interval is vertical, the run is zero and the gradient of the interval is not defined. This is shown by interval AB. If the interval is horizontal, the rise is zero as shown by interval PQ. The gradient of the interval is zero. The gradient of a line is defined to be the gradient of any interval within the line.

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This definition depends on the fact that two intervals on a line have the same gradient. Suppose AB and PQ are two intervals on the same straight line. Find its gradient. A line passes through the point 5, 7 and has gradient. Find the gradient of the line passing through a , b and 0, c.

The x -intercept of a line is the point at which it crosses the x -axis. The y -intercept of a line is the point at which it crosses the y -axis. In the diagram to the left the y -intercept is at A and the x -intercept at B. The second diagram shows a line parallel to the x -axis and it has a y -intercept at C. The third diagram shows a line parallel to the y -axis and it has an x -intercept at D. Can we find the equation of the line given suitable geometric information about the line?

The following shows that this can be done given the gradient of the line and the y -intercept. This passes through the point A 0, 2. This is called the equation of the line. Consider the line with gradient m and y -intercept c. If passes through the point A 0, c.