CHAPTER 1 · RESOURCES

CONNECT — Concept Note

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A concise revision of the approved lesson. For discovery, interaction and guided reasoning, return to the lesson.

CONCEPT

Distance Between Any Two Points

H = (x₂, y₁) shares a row with A and a column with B.
Signed changes are Δx = x₂ − x₁ and Δy = y₂ − y₁. Leg lengths are |Δx| and |Δy|; squaring removes the signs.
AB² is the sum of the squares of the two leg lengths. Take the non-negative square root for AB.

${AB}^{2} = {(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}$

$AB = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

Find horizontal change, find vertical change, square both, add, then take the square root. Use the diagram to check whether the result is reasonable.

One interval = one unit. AH is horizontal; HB is vertical. AB joins the endpoints directly.

A(-2, -1), B $(1, 3)$ . Horizontal change 3; vertical change 4. AB = 5 units.

Signed changes: Δx = 3, Δy = 4.
Leg lengths:
AH = | $x_{2}$ − $x_{1}$ | = 3
HB = | $y_{2}$ − $y_{1}$ | = 4
${AB}^{2}$ = 9 + 16 = 25

Find AB for A(−2, −1) and B(1, 3).

AB = 5 units. It exceeds either leg but is less than their sum, 7.

Adding the two leg lengths measures the route through the helper point, not the straight segment.

Why is 3 + 4 not the straight-line distance in this example?

Check your reasoning

7 measures the two-leg route. The direct hypotenuse has length √(9 + 16) = 5.