91Ó°ÊÓ

To find the distance between two points on a coordinate plane, you use the distance formula. This formula is derived from the Pythagorean theorem and it calculates the straight line distance between two points. The formula is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] where \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of the two points. Breakdown of the formula:

• \( x_2 - x_1 \) and \( y_2 - y_1 \) represent the differences in the x-coordinates and y-coordinates.
• Squaring these differences ensures they are positive.
• Adding the squares gives a number representing the area of the squares on the sides of the right triangle.
• Taking the square root converts this area back to a distance.

In our exercise, substituting the coordinates \( (4, 3) \) and \( (4, f) \) into this formula leads to: \[ 10 = \sqrt{(4 - 4)^2 + (f - 3)^2} \]. Notice that \( x_1 \) and \( x_2 \) are the same, which simplifies the formula.

When we applied the distance formula in the exercise, we ended up with the equation: \[ 10 = |f - 3| \]. This is an absolute value equation. Absolute value represents the distance between a number and zero on the number line, disregarding the direction. It is always non-negative. The equation \( 10 = |f - 3| \) means there are two possible cases:

• The quantity inside the absolute value is positive or zero: \( f - 3 = 10 \)
• The quantity inside the absolute value is negative: \( f - 3 = -10 \)

Therefore, solving it, we get two solutions:

• \( f - 3 = 10 \Rightarrow f = 13 \)
• \( f - 3 = -10 \Rightarrow f = -7 \)

This is because both 13 and -7 are exactly 10 units away from 3 on the number line.

The coordinate plane is a two-dimensional surface where each point is defined by a pair of numbers, commonly known as coordinates. The coordinate plane is formed by two number lines intersecting at a right angle:

• The horizontal number line is the x-axis.
• The vertical number line is the y-axis.

The origin, where both axes intersect, has coordinates (0,0). Each point on the plane can be described using an ordered pair \( (x, y) \), where:

• \( x \) is the position along the horizontal axis.
• \( y \) is the position along the vertical axis.

In our problem, the coordinates \( (4,3) \) and \( (4, f) \) denote two specific points on this plane. The distance formula is used to find how far apart these points are. By identifying the coordinates correctly and applying the correct formula, we can solve for unknown variables, gaining a deeper understanding of coordinate geometry.