Question Statement

Which sets of lines are concurrent in the given figures? Also identify and state the point of concurrency for each set.

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Background and Explanation

Concurrent lines are defined as three or more straight lines that intersect at a single common point. The specific point where these multiple lines meet is called the point of concurrency. To identify concurrent lines, we look for locations in a figure where multiple lines pass through the same exact point.

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Solution

Figure (i)

In this figure, we identify two distinct points where multiple lines intersect:

At point $A$: Three lines pass through this point. Therefore, lines $p$, $q$, and $m$ are concurrent.

• Point of concurrency: $A$
• Concurrent lines: $p$, $q$, and $m$

At point $B$: Three lines pass through this point. Therefore, lines $\ell$, $p$, and $n$ are concurrent.

• Point of concurrency: $B$
• Concurrent lines: $\ell$, $p$, and $n$

Note that line $p$ passes through both points $A$ and $B$, making it part of two different sets of concurrent lines at different locations.

Figure (ii)

In this figure, we identify two distinct points where more than two lines intersect:

At point $A$: Four lines pass through this point. Therefore, lines $a$, $b$, $c$, and $r$ are concurrent.

• Point of concurrency: $A$
• Concurrent lines: $a$, $b$, $c$, and $r$

At point $D$: Three lines pass through this point. Therefore, lines $p$, $q$, and $r$ are concurrent.

• Point of concurrency: $D$
• Concurrent lines: $p$, $q$, and $r$

Note that line $r$ passes through both points $A$ and $D$, participating in both sets of concurrent lines.

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Key Formulas or Methods Used

• Definition of concurrent lines: Three or more lines are concurrent if they all pass through a single common point
• Point of concurrency: The specific intersection point shared by concurrent lines
• Visual inspection method: Counting the number of lines passing through each intersection point in the geometric figure

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Summary of Steps

1. Examine the figure to locate all points where lines intersect.
2. At each intersection point, count how many lines pass through that specific point.
3. Identify any point where three or more lines meet as a point of concurrency.
4. List the names of all lines passing through that point to define the set of concurrent lines.
5. Repeat for all such points in the figure (there may be multiple points of concurrency in one diagram).