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)
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)
✨ Free AI Tools for You
Powered by Chatterbot AI
Answer
:
The correct option is A: (2, 1).
Given the equations of the sides of the triangle:
 $x + y  5 = 0$
 $x  y + 1 = 0$
 $y  1 = 0$
To identify the coordinates of the circumcentre:

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 $(2, 3)$.
 Solving these two equations simultaneously:

Check for rightangled triangle:
 The given triangle is rightangled, which means one of its angles is 90°.

Determine the hypotenuse: The hypotenuse is the side opposite the right angle. For a rightangled triangle, the circumcentre is the midpoint of the hypotenuse.

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).
India's 1^{st} AI Doubt Solver for CBSE, JEE, and NEET
Ask a Question for Freeand then it's just ₹212 a month