Exercises Coordinate Geometry 1

...

Solution to 5

(a) To show that all four sides are equal in length, we need to calculate their distances (length), using the Distance Formula.

d = √(changing in x)² + (changing in y)² d = √(x2 − x1)² + (y2 − y1)²

1. Distance between A(− 3, 2) and B(4, 3)
d_AB = √(4 − (− 3))² + (3 − 2)² = √7² + 1² = √50 = √2 · 5² = 5√2


2. Distance between B(4, 3) and C(9, − 2)
d_BC = √(9 − 4)² + (− 2 − 3)² = √5² + (−5)² = √50 = √2 · 5² = 5√2


3. Distance between C(9, − 2) and D(2, − 3)
d_CD = √(2 − 9)² + (− 3 − (− 2))² = √(−7)² + (−1)² = √50 = √2 · 5² = 5√2


4. Distance between D(2, − 3) and A(− 3, 2)
d_DA = √(2 − (−3))² + (− 3 − 2)² = √5² + (−5)² = √50 = √2 · 5² = 5√2

(b) As you can notice:
      - All for sides are equal in length
      - Opposite sides are parallel
but
      - The four angles are not right
      - The diagonals are not congruent

Therefore, the figure is still a quadrilateral, not a square, but a parallelogram.

Top ⤴

Solution to 6

(a) First solve the line eqaution 3x + 4y = 16 for y:
      that way you'll come to l₁: y = ⁻³/₄·x + 4, with the slope m₁ = ⁻³/₄.

      If l₂ passes through P and perpendicular to l₁, it means that the slope of m₂ ist negative       reciprocal to the slope of m₁
      ⇒ m₁ = ⁻¹/_m2 ⇔ m₂ = ⁻¹/_m1 = ⁻¹/_⁻³/4 = ⁴/₃, m₂ = ⁴/₃
      l₂ passes through P
      ⇒ 5 = ⁴/₃ − 7 + b, with b to be found!
      5 = ²⁸/₃ + b and b = ⁻¹³/₃

      Therefore, the equation of l₂: y = ⁴/₃·x − ¹³/₃

(b) Point of intersection of l₁ and l₂:
      this means l₁ = l₂
      ⇒ ⁻³/₄·x + 4 = ⁴/₃·x − ¹³/₃, solve for x:
      ⇒ x = 4
      then plot x into either l₁ or l₂:
      ⇒ y = ⁻³/₄·4 + 4 =
      In conclusion, the points l₁ and l₂ intersect at the point (4, 1)

(c) The perpendicular distance of P from the line l₁ represents the Hypotenuse of the slope       with the values x = 3 and y = 4
      therefore, we will use the Pythagoras formula for right triangle to find out the distance      concerned:

      so, d² = 3² + 4²    |   √ (square root it)
      ⇒ d = √3² + 4² = √5²
      ⇒ d = 5

Top ⤴

continue...