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Distance between two points · 3 marks

Plot point Z (5, −6) on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides.
(Comment: Answers may differ from person to person.)

Answer: One answer: N (5, −2) and I (2, −2), with the right angle at N. ZN = 4, NI = 3, IZ = √(32 + 42) = 5 units. (Other choices are also correct.)

Step-by-step solution

Given: Z (5, −6) in Quadrant IV

Idea: The easiest right angle is one between a vertical side and a horizontal side. Choose N straight above (or beside) Z, then I level with N. The two legs are differences of coordinates and the hypotenuse comes from the Baudhāyana–Pythagoras theorem.

xy−11234567−7−6−5−4−3−2−11Z (5, −6)N (5, −2)I (2, −2)345
  1. Plot Z (5, −6): 5 units right of O, then 6 units down. It is in Quadrant IV.½ mark
  2. Take N (5, −2), straight above Z, and I (2, −2), level with N. ZN is vertical and NI is horizontal, so the angle at N is a right angle.½ mark
  3. ZN = −2 − (−6) = 4 units; NI = 5 − 2 = 3 units.1 mark
  4. IZ is the hypotenuse: IZ = √(NI2 + ZN2) = √(32 + 42) = √25 = 5 units.1 mark
  5. Another valid choice: N (5, 0) and I (0, 0), giving sides 6, 5 and √61. Any triangle with a right angle is correct, as long as the lengths are worked out correctly.
With N (5, −2) and I (2, −2), triangle IZN is right-angled at N with ZN = 4 units, NI = 3 units and IZ = 5 units. Other correct triangles are possible.

Check: IZ by the distance formula: √((5 − 2)2 + (−6 − (−2))2) = √(9 + 16) = 5 ✓.

Answer to write in the exam

Z (5, −6) plotted; N (5, −2), I (2, −2)

ZN ∥ y-axis, NI ∥ x-axis ⇒ ∠INZ = 90°

ZN = −2 − (−6) = 4 units

NI = 5 − 2 = 3 units

IZ = √(32 + 42) = √25 = 5 units (Baudhāyana–Pythagoras theorem)

∴ Sides: 3 units, 4 units, 5 units

Common mistakes that cost marks

  • Plotting Z (5, −6) at 5 down and 6 right. The first coordinate is always the move along the x-axis.
  • Choosing three points without checking there is a right angle. Use one vertical and one horizontal side to be sure.
  • Writing ZN = −2 − 6 = −8. Take care with the sign: −2 − (−6) = −2 + 6 = 4.

How this can come in the exam

MCQ (1 mark)

Z (5, −6), N (5, −1) and I (2, −1) form a right-angled triangle. The length IZ is

  1. √34 units
  2. 8 units
  3. 34 units
  4. √8 units
Show answer

(A) √34 units
Legs ZN = 5 and NI = 3: IZ = √(9 + 25) = √34.

Short answer (2 marks)

Construct a right-angled isosceles triangle with its right angle at Z (5, −6) and legs of 4 units parallel to the axes. Give the other two vertices and the length of the hypotenuse.

Show answerVertices (5, −2) and (1, −6) (1 mark). Hypotenuse = √(42 + 42) = √32 = 4√2 units (1 mark).

Try one yourself

Plot Y (−2, 7). Using the point directly below Y on the x-axis and the origin, form a right-angled triangle and find its sides.

Show answer

Third vertex (−2, 0). Sides 7, 2 and √53 units (√(4 + 49)).

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