Question

The equations of the sides of a triangle are x + y - 5 = 0, x - y + 1 = 0, and y - 1 = 0. Then the coordinates of the circumcentre are:

(2, 1)

(1, 2)

(2, -2)

(1, -2)

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Answer

:

The correct option is A: (2, 1).

Given the equations of the sides of the triangle:

  1. $x + y - 5 = 0$
  2. $x - y + 1 = 0$
  3. $y - 1 = 0$

To identify the coordinates of the circumcentre:

  1. Calculate the vertices:

    • From $y - 1 = 0$, we get $y = 1$.
    • Substituting $y = 1$ in the other equations:
      • $x + 1 - 5 = 0 \implies x = 4$. Thus, one vertex is $(4, 1)$.
      • $x - 1 + 1 = 0 \implies x = 0$. Thus, another vertex is $(0, 1)$.
    • The intersection of $x + y - 5 = 0$ and $x - y + 1 = 0$:
      • Solving these two equations simultaneously:
        • Adding these two equations: $(x + y - 5) + (x - y + 1) = 0 \implies 2x - 4 = 0 \implies x = 2$
        • Substituting back: $2 + y - 5 = 0 \implies y = 3$. Thus, the third vertex is $(2, 3)$.
  2. Check for right-angled triangle:

    • The given triangle is right-angled, which means one of its angles is 90°.
  3. Determine the hypotenuse: The hypotenuse is the side opposite the right angle. For a right-angled triangle, the circumcentre is the midpoint of the hypotenuse.

  4. Find the midpoint of the hypotenuse:

    • The vertices forming the right angle are $(0, 1)$ and $(2, 3)$, and the third vertex is $(4, 1)$.
    • Hypotenuse is between $(4, 1)$ and $(0, 1)$.
    • Midpoint of hypotenuse: $$ \left( \frac{4 + 0}{2}, \frac{1 + 1}{2} \right) = (2, 1) $$

Thus, the coordinates of the circumcentre are (2, 1).


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