Before you read

Answer first — see what you already know.

What is the midpoint of $(4, - 2)$ and $(- 6, 8)$ ?

A
$(- 1, 3)$
B
$(5, 5)$
C
$(- 2, 6)$
D
$(1, - 3)$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

The midpoint of $A (1, 5)$ and $B (x, y)$ is $(4, 3)$ . What is $B$ ?

A
$(7, 1)$
B
$(3, - 2)$
C
$(2.5, 4)$
D
$(8, 6)$

Check your answer

Answer:

Theory

Midpoint and Distance Formulas

Midpoint of $(x_1, y_1)$ and $(x_2, y_2)$ :
$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Midpoint M = ((1+5)/2, (2+6)/2) = (3, 4)

Distance between two points:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Distance = √((4-(-2))² + (5-1)²) = √(52) ≈ 7.2

The distance formula IS the Pythagorean theorem applied to coordinates.

Example 1

Find the midpoint and distance between $(2, 3)$ and $(8, 11)$ .

Midpoint: $\left(\frac{2+8}{2}, \frac{3+11}{2}\right) = (5, 7)$

Distance: $\sqrt{6^2 + 8^2} = \sqrt{100} = 10$

Midpoint

(5, 7)

, Distance

= 10

Tip

ACT Pro Tip

If the ACT asks for the midpoint, it's just averaging the coordinates. If it gives you a midpoint and one endpoint and asks for the other, set up: $\frac{x_1 + x_2}{2} = M_x$ and solve.

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