CONNECT · CHAPTER 1

CONNECT — Distance Between Any Two Points

Think first. Use AI purposefully. Verify always.

01 · DISCOVER

Start with a question.

Can a diagonal journey be shorter than travelling along two sides of a right triangle?

Make a prediction before exploring. You do not need a rule yet.

02 · UNDERSTAND

Distance Between Any Two Points

Join A to a helper point H directly across, then move vertically from H to B. The magnitudes of the two perpendicular changes are the leg lengths of a right triangle. AB is its hypotenuse. Connect this picture to the Baudhāyana–Pythagoras theorem before using a formula.

03 · KEY IDEAS

Key Ideas

H = ( $x_{2}$ , $y_{1}$ ) shares a row with A and a column with B.
Signed changes are Δx = $x_{2}$ − $x_{1}$ and Δy = $y_{2}$ − $y_{1}$ . Leg lengths are |Δx| and |Δy|; squaring removes the signs.
${AB}^{2}$ is the sum of the squares of the two leg lengths. Take the non-negative square root for AB.

04 · VISUALISE

Explore the idea.

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

Explore, then verify

Point Ax: -2y: -1

Point Bx: 1y: 3

Drag a point, or focus a slider and use arrow keys.

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

05 · METHOD

From understanding to a method.

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

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

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

06 · ONE WORKED EXAMPLE

Follow each step.

Find AB for A $(- 2, - 1)$ and B $(1, 3)$ .

Choose H $(1, - 1)$ . AH = 3 and HB = 4.
${AB}^{2}$ = $3^{2}$ + $4^{2}$ = 9 + 16 = 25.
AB = 5 units. It exceeds either leg but is less than their sum, 7.

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

Read the matching diagram

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

07 · AIMS METHOD

Your turn to reason.

Attempt independently
Investigate the idea
Make connections
Scrutinise your reasoning

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

Your reasoning

Your written reasoning is not automatically graded or sent anywhere.

08 · AIMS PRACTICE SPRINT

Vary the situation. Keep the idea.

Work through 10 questions on this concept. Select an answer before checking it; use the feedback to revise your reasoning.

Start practice sprint

09 · CHECK & FEEDBACK

Correct answers receive an explanation. Incorrect answers receive a hint and another attempt. This is practice, not an examination.

Review status is self-reported, not a mastery score. Progress is stored only in this browser; no account or tracking service is used.

TEXTBOOK CONNECTION · CONNECT

Now connect with your textbook.

Choose one item. Try it independently, then use an AIMS hint or compare the reasoning.

Textbook Guided Example

Figs. 1.6–1.9 →

End-of-Chapter Practice

End-of-Chapter Q4 →

★ AIMS Think+

AT+ · Explore deeper. Reason beyond the routine.

End-of-Chapter ★ Q6 →
End-of-Chapter ★ Q7 →
End-of-Chapter ★ Q12 →Also connects to APPLY

Original AIMS explanations linked to verified textbook references. Completion remains untracked.