Chapter 11: Loci and Construction Definitions & Formulas

① Define Locus.

A locus (plural loci) is a set of points that follow a given rule. In geometry, loci help define the positions of points relative to one another or to geometric figures.

For example, if a point moves so that it always stays 5 cm away from a fixed point, then the locus of that moving point is a circle with a radius of 5 cm, centered at the fixed point.

② Write types of triangles.

Types of Triangles (by sides):

• Scalene triangle: All sides are of different lengths.
• Isosceles triangle: Two sides are of equal length.
• Equilateral triangle: All sides are of equal length.

Types of Triangles (by angles):

• Acute-angled triangle: All angles measure less than \(90^\circ\).
• Obtuse-angled triangle: One angle measures greater than \(90^\circ\).
• Right-angled triangle: One angle measures exactly \(90^\circ\).

③ What is the Triangle Inequality Theorem?

The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

For example, for a triangle with sides 5 cm, 7 cm, and 8 cm:

\(5 + 7 > 8\)

\(5 + 8 > 7\)

\(7 + 8 > 5\)

So, the triangle satisfies the triangle inequality rule.

④ Define perpendicular bisector.

A perpendicular bisector is a line that intersects a line segment at right angle and divides it into two equal parts.

In other words, it intersects the line segment at its midpoint and forms right angles \(90^\circ\) with it.

⑤ Define median of a triangle.

A median of a triangle is a line segment that joins a vertex to the midpoint of the side that is opposite to that vertex.

⑥ What is centroid?

The point of concurrency of the medians of a triangle is called centroid of the triangle.

⑦ What is a point of concurrency?

A point of concurrency is the single point where three or more lines, rays, or line segments intersect or meet in a geometric figure.

⑧ What is circumcentre?

The point of concurrency of perpendicular bisectors of the sides of a triangle is called circumcentre.

⑨ What is an angle bisector of a triangle?

An angle bisector of a triangle is a line or ray that divides an angle into two equal parts, creating two smaller angles that are congruent (each having half the measure of the original angle).

⑩ What is an altitude of a triangle?

Altitude is a ray drawn perpendicular from a vertex to the opposite side of the triangle.

There are three altitudes of the triangle which meet at a single point, i.e. the altitudes of a triangle are concurrent.

⑪ What is orthocentre?

The point of concurrency of the altitudes of the triangle is called the orthocentre of the triangle.

⑫ Define circle as a locus.

The locus of a point whose distance is constant from a fixed point is called a circle.

For example, the locus of a point \( P \) whose distance is \( 3\ \text{cm} \) from a fixed point \( O \) is a circle of radius \( 3\ \text{cm} \) and centre at \( O \).

⑬ Define parallel lines as a locus.

The locus of a point whose distance from a fixed line is constant are parallel lines (i.e. \( l \) and \( m \)).

For example, the locus of a point \( P \) whose distance is \( 2.5\ \text{cm} \) from a fixed line \( AB \) forms two parallel lines.

The locus of points equidistant from a line segment forms a long oval-like shape around the line — like a sausage or a running track.

⑭ Define perpendicular bisector as a locus.

The locus of a point whose distance from two fixed points is constant is called a perpendicular bisector.

For example, the locus of a point \( P \) equidistant from \( A \) and \( B \) is the perpendicular bisector of line segment \( AB \).

⑮ Define angle bisector as a locus.

The locus of a point whose distance is constant from two intersecting lines is called an angle bisector.

For example, the locus of a point \( P \) equidistant from lines \( AB \) and \( CD \) (intersecting at \( O \)) is the angle bisector \( l \) of \( \angle AOC \) and \( \angle BOD \).