The perpendicular bisectors of the sides of an acute-angled triangle are drawn. Which of these statements is correct? They intersect

Analyze the question: The perpendicular bisectors of a triangle's sides meet at one point called the circumcentre, the centre of the circle passing through all three vertices (the circumcircle). The question asks where this point lies for an acute-angled triangle.

Concept: The position of the circumcentre depends on the type of triangle.

| Type of triangle | Position of circumcentre | |---|---| | Acute-angled | Inside the triangle | | Right-angled | On the midpoint of the hypotenuse | | Obtuse-angled | Outside the triangle |

Why this is so:

• The circumcentre lies on the circumcircle's diameter-related geometry. In a right-angled triangle, the angle in a semicircle is 90°, so the hypotenuse is a diameter, and the centre is its midpoint.
• If the triangle is acute, every angle is less than 90°. Each side then subtends an inscribed angle below 90°, so the centre lies on the same side of each side as the opposite vertex. That places it inside the triangle.
• If one angle is obtuse, the centre falls on the opposite side of the longest side from the obtuse vertex, so it lies outside.

Checking the options:

• A. On a vertex: impossible, since the circumcentre is equidistant from the vertices and the distance would be zero.
• B. At the midpoint of a side: this happens only for a right-angled triangle.
• C. Inside the triangle: correct for an acute-angled triangle.
• D. Outside the triangle: this happens only for an obtuse-angled triangle.

Answer: C. Inside the triangle

Exam tip: Remember it as "Acute = Inside, Right = On the side, Obtuse = Outside." The same rule applies to the orthocentre (the meeting point of the altitudes). For the centroid and incentre, the point is always inside.