Understanding the Triangle and Angle Bisector Problem

The question describes a triangle ΔXYZ with given side lengths XY = 4 cm and XZ = 5 cm. A point Q is located on the side YZ such that the line segment XQ bisects the angle ∠X. We are asked to find the ratio of the lengths YQ and QZ, i.e., YQ ∶ QZ.

This problem can be solved using a fundamental theorem in geometry known as the Angle Bisector Theorem.

Applying the Angle Bisector Theorem

The Angle Bisector Theorem states that in a triangle, if a line segment bisects an angle and intersects the opposite side, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

In our case, XQ is the angle bisector of ∠X, and it intersects the opposite side YZ at point Q. According to the Angle Bisector Theorem, the ratio of the lengths of the two segments YQ and QZ must be equal to the ratio of the lengths of the other two sides of the triangle, XY and XZ.

Mathematically, the theorem states:

\( \frac{\text{YQ}}{\text{QZ}} = \frac{\text{XY}}{\text{XZ}} \)

Calculating the Ratio YQ ∶ QZ

We are given the following side lengths:

• XY = 4 cm
• XZ = 5 cm

Now, we can substitute these values into the Angle Bisector Theorem formula:

\( \frac{\text{YQ}}{\text{QZ}} = \frac{4 \text{ cm}}{5 \text{ cm}} \)

The units (cm) cancel out, leaving us with the ratio:

\( \frac{\text{YQ}}{\text{QZ}} = \frac{4}{5} \)

This means that the ratio YQ ∶ QZ is 4 ∶ 5.

Summary of the Steps

Here is a quick summary of the steps taken to find the ratio:

1. Identify the triangle and the angle bisector.
2. Recall and apply the Angle Bisector Theorem.
3. Substitute the given side lengths into the theorem's formula.
4. Calculate the resulting ratio.

Checking the Options

Let's compare our calculated ratio with the given options:

Our calculated ratio YQ ∶ QZ is 4 ∶ 5, which matches Option 4.

Revision Table: Key Concepts

Additional Information: The Angle Bisector Theorem in Depth

The Angle Bisector Theorem is a crucial result in triangle geometry. It provides a direct relationship between the lengths of the sides of a triangle and the segments created by an angle bisector intersecting the opposite side.

• Theorem Statement: For a triangle ABC, if AD is the angle bisector of ∠A, where D is on BC, then \( \frac{BD}{DC} = \frac{AB}{AC} \).
• Converse: The converse of the theorem is also true. If a point D on side BC of a triangle ABC divides BC in the ratio AB : AC, i.e., \( \frac{BD}{DC} = \frac{AB}{AC} \), then AD is the angle bisector of ∠A.
• Proof Sketch: The theorem can be proven by extending one side of the triangle (say, AB) to a point E such that CE is parallel to AD. Using properties of parallel lines and corresponding/alternate angles, one can show that triangle ACE is isosceles with AC = AE, and then apply the property of similar triangles (specifically, triangle BAD and triangle BCE).
• Applications: This theorem is widely used in solving geometry problems involving ratios, lengths of segments, and proofs related to angle bisectors.