Three parallel lines x, y and z are cut by two transversals m and n. Transversal m cuts the lines, x, y, z at P, Q, R respectively, and transversal n cuts the lines x, y, z at L, M, N respectively. If PQ = 3 cm, QR = 9 cm and MN = 10.5cm, then what is the length of LM?

Question

Prepp Team · Accepted Answer

Understanding the Problem: Parallel Lines and Transversals

The question describes a geometric setup involving three parallel lines, labelled x, y, and z. These lines are intersected by two other lines, called transversals, labelled m and n. Transversal m cuts the parallel lines at points P (on x), Q (on y), and R (on z). Transversal n cuts the same parallel lines at points L (on x), M (on y), and N (on z).

We are given the lengths of certain segments formed on these transversals by the parallel lines:

On transversal m: PQ = 3 cm and QR = 9 cm.
On transversal n: MN = 10.5 cm.

We need to find the length of the segment LM on transversal n.

Applying the Intercept Theorem (Property of Parallel Lines and Transversals)

This problem can be solved using a fundamental theorem in geometry related to parallel lines and transversals. The theorem states that if three or more parallel lines cut off segments on a transversal, then they cut off proportional segments on any other transversal.

In our case, the three parallel lines x, y, and z are cut by transversals m and n. According to the theorem, the ratio of the segments on transversal m must be equal to the ratio of the corresponding segments on transversal n.

The segments on transversal m are PQ and QR. The corresponding segments on transversal n are LM and MN. Therefore, we can set up the following proportion:

\[ \frac{\text{Length of segment on m between x and y}}{\text{Length of segment on m between y and z}} = \frac{\text{Length of segment on n between x and y}}{\text{Length of segment on n between y and z}} \]

Substituting the segment names:

\[ \frac{PQ}{QR} = \frac{LM}{MN} \]

Solving for the Unknown Length LM

We are given the values for PQ, QR, and MN. Let's substitute these values into the proportion:

Segment	Transversal	Given Length
PQ	m	3 cm
QR	m	9 cm
MN	n	10.5 cm
LM	n	? (To find)

The equation becomes:

\[ \frac{3 \text{ cm}}{9 \text{ cm}} = \frac{LM}{10.5 \text{ cm}} \]

Now, we can solve for LM. First, simplify the fraction on the left side:

\[ \frac{3}{9} = \frac{1}{3} \]

So the equation is:

\[ \frac{1}{3} = \frac{LM}{10.5} \]

To find LM, we can multiply both sides of the equation by 10.5:

\[ LM = \frac{1}{3} \times 10.5 \]

\[ LM = \frac{10.5}{3} \]

Performing the division:

\[ LM = 3.5 \]

So, the length of LM is 3.5 cm.

Summary of Calculation Steps

Identify the parallel lines and transversals.
Recognize the segments formed on each transversal.
Apply the Intercept Theorem, stating that the ratio of corresponding segments on the transversals is equal: \( \frac{PQ}{QR} = \frac{LM}{MN} \).
Substitute the given values: \( \frac{3}{9} = \frac{LM}{10.5} \).
Simplify the equation: \( \frac{1}{3} = \frac{LM}{10.5} \).
Solve for LM: \( LM = \frac{1}{3} \times 10.5 = 3.5 \).

The length of LM is 3.5 cm.

Revision Table: Key Concepts

Concept	Description	Relevance to Problem
Parallel Lines	Lines in a plane that never meet.	x, y, and z are parallel lines, crucial for the theorem.
Transversal	A line that intersects two or more other lines.	m and n are transversals, forming segments.
Intercept Theorem	Parallel lines cut proportional segments on any transversal.	Used to set up the equation \( \frac{PQ}{QR} = \frac{LM}{MN} \).
Proportion	An equation stating that two ratios are equal.	The theorem leads to a proportion, which is solved.

Additional Information: Extensions of the Intercept Theorem

The Intercept Theorem is a powerful result. Here are some related ideas:

Basic Proportionality Theorem (BPT): Also known as Thales's Theorem, BPT is a special case of the Intercept Theorem involving a triangle and a line parallel to one of its sides. If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides proportionally. You can visualize the parallel lines x, y, z as part of a figure where transversals m and n might converge to form a triangle (or extend beyond to form similar triangles).
Midpoint Theorem: This is a specific application where the parallel lines cut off equal segments on one transversal. If a line segment joins the midpoints of two sides of a triangle, then it is parallel to the third side and is half the length of the third side. This implies that if the parallel lines cut equal intercepts on one transversal, they cut equal intercepts on any other transversal as well.
Applications: The Intercept Theorem is used in various geometric constructions and proofs, and it's fundamental for understanding similarity in triangles.

Understanding how parallel lines divide transversals proportionally is key to solving this type of geometry problem.