Geometry Lesson Plan - Intercept Theorem

The intercept theorem is an important elementary geometry theorem. It is very useful for solving the geometric problems related to the ratios of various line segments.

The theorem states that if two or more parallel lines are intersected by two self intersecting lines, then the ratios of the line segments of the first intersecting line is equal to the ratio of the similar line segments of the second intersecting line. Bit confused? Ok, let’s explain it using the following example:

 

In the figure above, two parallel lines FI and JM is cut by two self intersecting lines AE and ON. Two triangles are formed as a result of the intersections of the lines. According to the intercept theorem, the ratio of the line segments created from ON should be equal to the ratio of the lengths of the line segments created from AE. Therefore,

|BG|: |GK| = |BH|:|HL|

• From the above figure, it is clear that the ΔBGH and ΔBKL are similar.
• From the rules of similar triangle, we can write:

BK/BG = BL/BH………….eqn.1

• Now, we can write, BK=BG+GK and BL=BH+HL
• So, from the eqn.1 we can get:

(BG+GK)/BG = (BH+HL)/BH

Or, 1+ (GK/BG) = 1+ (HL/BH)

Or, GK/BG = HL/BH

Or, |BG|: |GK| = |BH|:|HL|

• Explain the concept of the intercept theorem.
• Explain the proof of the concept with an example.
• Draw different sets of parallel lines and the lines intersecting the parallel lines and ask the students to write the proportionality of the different lines segments.

• Give the following exercise:

Problem: If the points B, D and F are the mid-points of the lines AC, CE and AE respectively, then prove that:

Area of ΔAFB = Area of ΔFED

 

Solution:

• Since the points F and B are the mid-points of the lines AE and AC respectively, so:

AF/FE = AB/BC

• Hence, the lines BF and EC must be parallel (according to the intercept theorem).

• Similarly, the lines BD and AE are parallel.

• So, for the quadrilateral BDEF all the opposite sides are parallel to each other. Hence, the quadrilateral BDEF is a parallelogram with ED as one of the diagonal.

Hence, Area of ΔEFD = Area of ΔBDF……….eqn2

• Similarly, the quadrilateral ABDF is also a parallelogram with BF as one of the diagonal.

Hence, Area of ΔABF = Area of ΔBDF………eqn3

• From the Eqn.2 &3, it can be prove that:

Area of ΔAFB = Area of ΔFED

The geometry lesson plan on the intercept theorem will help students understand the important concept of the theorem. The geometry theorem proof of the concept will help implementing the theorem for different geometric problems.