What is the Side Splitter Theorem? A Complete Introduction and Exploration

Step-by-step Guide: the Side Splitter Theorem

Understanding the Basics:
The Side Splitter Theorem comes into play when a line (or a segment) is drawn parallel to one side of a triangle, intersecting the other two sides. This line divides the two sides of the triangle into segments that are proportional.

Mathematical Statement of the Theorem:
Let’s assume triangle \(ABC\) has a line segment \(DE\) parallel to side \(AC\) and intersecting \(AB\) and \(BC\) at \(D\) and \(E\) respectively. Then, according to the Side Splitter Theorem:
\( \frac{BD}{DA} = \frac{BE}{EC} \)

Proof of the Side Splitter Theorem:
This theorem can be proved using similar triangles. Since \(DE\) is parallel to \(AC\) and \(AC\) is transversal, angles \(ADE\) and \(BAC\) are alternate interior angles and thus congruent. Similarly, angles \(BDE\) and \(BCA\) are congruent. Hence, triangles \(ADE\) and \(ABC\) are similar by the Angle-Angle (AA) similarity postulate. This means that the ratio of their corresponding sides is equal, leading to the above-stated proportion.

Examples

Example 1:
In triangle \(XYZ\), a line parallel to \(YZ\) intersects \(XY\) and \(XZ\) at points \(P\) and \(Q\) respectively. If \(XP\) is \(3 \text{ cm}\), \(PY\) is \(9 \text{ cm}\), and \(XQ\) is \(4 \text{ cm}\), find the length of \(QZ\).

Solution:
Using the Side Splitter Theorem:
\( \frac{XP}{PY} = \frac{XQ}{QZ} \)
Plugging in the known values:
\( \frac{3 \text{ cm}}{9 \text{ cm}} = \frac{4 \text{ cm}}{QZ} \)
Simplifying:
\( \frac{1}{3} = \frac{4 \text{ cm}}{QZ} \)
Therefore, \(QZ = 12 \text{ cm}\).

Example 2:
In triangle \(MNO\), a line segment \(RS\) is parallel to \(MO\) and intersects \(MN\) and \(NO\) at \(R\) and \(S\) respectively. Given that \(MR = 5 \text{ cm}\), \(RN = 10 \text{ cm}\), and \(NS = 8 \text{ cm}\), find the length of \(SO\).

Solution:
By the Side Splitter Theorem:
\( \frac{MR}{RN} = \frac{SO}{NS} \)
Plugging in the given values:
\( \frac{5 \text{ cm}}{10 \text{ cm}} = \frac{SO}{8 \text{ cm}} \)
Simplifying:
\( \frac{1}{2} = \frac{SO}{8 \text{ cm}} \)
Thus, \(SO = 4 \text{ cm}\).

Practice Questions:

1. In triangle \(ABC\), line segment \(DE\) is parallel to \(BC\). If \(BD = 6 \text{ cm}\), \(DA = 18 \text{ cm}\), and \(AE = 9 \text{ cm}\), find \(CE\).
2. Given triangle \(PQR\), line segment \(ST\) is parallel to \(QR\), \( ST \) divides \(PR\) into \(PT\) and \(TR\) in the ratio 2:3. If \(QR = 15 \text{ cm}\), find \(ST\).

Answers:

1. \(CE = 3 \text{ cm}\)
2. \(ST = 6 \text{ cm}\)