Find the coordinates of a point on the x-axis whi

To find the coordinates of a point on the x-axis that is equidistant from the points (1, 2) and (5, 6), we can denote the point on the x-axis as (x, 0).

Step 1: Calculate Distances

We need to set the distances from (x, 0) to both (1, 2) and (5, 6) equal to each other.

Distance Formula

The distance between two points (x1, y1) and (x2, y2) is given by:

D = √((x2 - x1)² + (y2 - y1)²)

Distance to (1, 2)

Using the distance formula:

D1 = √((x - 1)² + (0 - 2)²) = √((x - 1)² + 4)

Distance to (5, 6)

Similarly, for the point (5, 6):

D2 = √((x - 5)² + (0 - 6)²) = √((x - 5)² + 36)

Step 2: Set Distances Equal

Now, we set D1 equal to D2:

√((x - 1)² + 4) = √((x - 5)² + 36)

Step 3: Square Both Sides

To eliminate the square roots, we square both sides:

(x - 1)² + 4 = (x - 5)² + 36

Step 4: Expand and Simplify

Expanding both sides gives:

• (x² - 2x + 1 + 4) = (x² - 10x + 25 + 36)

Which simplifies to:

x² - 2x + 5 = x² - 10x + 61

Step 5: Rearranging the Equation

Subtract x² from both sides:

-2x + 5 = -10x + 61

Now, rearranging gives:

8x = 56

Step 6: Solve for x

Dividing both sides by 8 results in:

x = 7

Final Coordinates

The coordinates of the point on the x-axis that is equidistant from (1, 2) and (5, 6) are:

(7, 0)