Loci • Topic 3 of 3

Locus of a point equidistant from two intersecting lines

What is the locus of a point equidistant from two intersecting lines? The locus of a point equidistant from two intersecting lines is a pair of straight lines that slice exactly through the middle of the angles formed where those lines cross. In geometry, this path is called an angle bisector.

When two straight lines cross each other, they form four internal angles opposite each other in pairs (two matching acute angles and two matching obtuse angles). Because there are two distinct sets of opening gaps, the complete locus consists of two perpendicular lines crossing through the same intersection vertex. Each line splits one pair of vertically opposite angles into two equal halves.

Let us explore some simple real-world examples:

  • A corner walkway: Imagine two straight fences meeting at a street corner to form a corner angle. If you want to lay down a stone path so that a person walking on it is always an equal distance from both fences, you must walk precisely along the line that divides that corner angle in half.
  • An airport runway approach: Two straight flight paths intersect near a control tower. A safety guide beam is projected along the angle bisector line so that aircraft stay safely equidistant from both flight paths.

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DIAGRAM 1: THE ANGLE BISECTOR PATHS

                             Locus Line 1 (Angle Bisector)
                                    /
                     Line 1        /
                           \      /
                            \    /  P (Equal distances to both lines)
                             \  / .
                              \/_ _ _ _ _ Distance d
                              /\
                             /  \
                            /    \
                           /      \
                     Line 2        \
                                    \
                               Locus Line 2

DIAGRAM 2: THE FOUR-WAY INTERSECTION SPLIT

                              Locus 1
                                 |  / Line 1
                                 | /
                     _ _ _ _ _ _ |/_ _ _ _ _ _ Locus 2
                                /|
                               / |
                        Line 2/  |
Solved Examples
7
Worked Example
Two straight pathways intersect at a central junction point V, forming an opening angle of 60 degrees between them. A park sprinkler moves along a path keeping an equal distance from both pathways. Describe its locus path and calculate the two split angles created by this path at the junction.
Solution
  1. Step 1: Identify the locus shape from the description.*
  2. A point staying equidistant from two intersecting straight lines forms the angle bisectors of those lines.*
  3. Step 2: Calculate the split angle for the internal region.*
  4. The main angle bisector will divide the given 60-degree angle into two equal parts.*
  5. Split Angle = 60 degrees / 2 = 30 degrees.*
  6. Answer: The path is the angle bisector of the intersecting lines, creating two split angles of 30 degrees each.
8
Worked Example
In a triangular garden patch ABC, a gardener wants to install a water fountain at a location that is completely equidistant from side fence AB and side fence AC. Describe the geometric line along which the fountain must be positioned.
Solution
  1. Step 1: Translate the side fences into geometric elements.*
  2. Side fence AB and side fence AC are two straight lines that meet and intersect at vertex A.*
  3. Step 2: Apply the intersecting lines locus rule.*
  4. The locus of points equidistant from two intersecting lines is the bisector of the angle formed between them.*
  5. Therefore, the fountain must lie on the line that bisects angle BAC.*
  6. Answer: The fountain must be positioned along the internal angle bisector of angle A.
9
Worked Example
Two lines, L1 and L2, intersect at an origin point. The acute angle between them is 40 degrees. Find the measure of the angle formed between the two separate locus lines that represent the complete equidistant path for this intersection.
Solution
  1. Step 1: Identify the components of the complete locus.*
  2. The complete locus consists of two angle bisectors: one bisecting the acute angle pair and one bisecting the obtuse angle pair.*
  3. Step 2: Find the values of the intersecting angles.*
  4. Acute angle = 40 degrees.*
  5. Obtuse angle = 180 degrees - 40 degrees = 140 degrees.*
  6. Step 3: Calculate the half-angles created by the bisectors.*
  7. Half of the acute angle = 40 / 2 = 20 degrees.*
  8. Half of the obtuse angle = 140 / 2 = 70 degrees.*
  9. Step 4: Sum the adjacent half-angles to find the angle between the two locus lines.*
  10. Angle between locus lines = 20 degrees + 70 degrees = 90 degrees.*
  11. Note: The bisectors of any pair of adjacent supplementary angles are always perpendicular to each other!*
  12. Answer: The angle formed between the two locus lines is 90 degrees.
  13. --

Key Points

  • The locus of a point equidistant from two intersecting lines is a pair of angle bisectors.
  • An angle bisector divides an interior angle into two perfectly equal angular sectors.
  • Every single point selected on an angle bisector is at an equal perpendicular distance from both lines.
  • Because crossing lines create both acute and obtuse gaps, the complete locus forms two separate lines.
  • These two resulting locus lines always intersect each other at a right angle of 90 degrees.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The locus of points equidistant from two intersecting lines is the:
Explanation: The two angle bisectors.
Q2.The two angle bisectors are ___ to each other.
Explanation: Mutually perpendicular.
Q3.Each bisector bisects the ___ between the lines.
Explanation: The angle.
Q4.The number of angle bisectors of two intersecting lines is:
Explanation: Two bisectors.
Practice more MCQs → 📝 Assignment · 35 marks