Solving the Triangle Geometry Problem: Finding BC

The problem asks us to find the length of side BC in a triangle ABC, given the lengths of segments on sides AB and AC, and the length of the segment PQ connecting the points on those sides.

We are given the following lengths:

• AP = 4 cm (P is on AB)
• PB = 6 cm (P is on AB)
• AQ = 5 cm (Q is on AC)
• QC = 7.5 cm (Q is on AC)
• PQ = 6 cm

First, let's find the total lengths of sides AB and AC:

• AB = AP + PB = 4 cm + 6 cm = 10 cm
• AC = AQ + QC = 5 cm + 7.5 cm = 12.5 cm

Now, let's look at the ratios of the corresponding segments on sides AB and AC. Consider the triangles APQ and ABC. They share the common angle \( \angle A \). We can compare the ratios of the sides adjacent to angle A:

• Ratio of AP to AB: \( \frac{AP}{AB} = \frac{4}{10} = \frac{2}{5} = 0.4 \)
• Ratio of AQ to AC: \( \frac{AQ}{AC} = \frac{5}{12.5} = \frac{50}{125} = \frac{2}{5} = 0.4 \)

Since \( \frac{AP}{AB} = \frac{AQ}{AC} = 0.4 \), and the angle \( \angle A \) is included between these sides in both triangles APQ and ABC, we can conclude that \( \triangle APQ \) is similar to \( \triangle ABC \) by the SAS (Side-Angle-Side) similarity criterion.

The ratio of similarity between \( \triangle APQ \) and \( \triangle ABC \) is 0.4 (or 2/5). This means that the ratio of any pair of corresponding sides in the two triangles is equal to this value.

The side PQ in \( \triangle APQ \) corresponds to the side BC in \( \triangle ABC \). Therefore, the ratio of their lengths must also be equal to the similarity ratio:

\( \frac{PQ}{BC} = \frac{AP}{AB} = \frac{AQ}{AC} \)

\( \frac{6}{BC} = 0.4 \)

To find BC, we can rearrange the equation:

\( BC = \frac{6}{0.4} \)

\( BC = \frac{6}{\frac{4}{10}} = 6 \times \frac{10}{4} = \frac{60}{4} = 15 \)

Thus, the length of BC is 15 cm.

Here is a summary of the lengths and ratios:

Revision Table: Triangle Similarity Criteria

Triangle similarity is a fundamental concept in geometry. Two triangles are similar if they have the same shape, which means their corresponding angles are equal and their corresponding sides are in proportion.

Here are the main criteria for proving triangle similarity:

• AA (Angle-Angle) Similarity: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
• SSS (Side-Side-Side) Similarity: If the corresponding sides of two triangles are in proportion, then the triangles are similar.
• SAS (Side-Angle-Side) Similarity: If two sides of one triangle are in proportion to two corresponding sides of another triangle, and the included angles are congruent, then the triangles are similar.

Additional Information: Properties of Similar Triangles

When two triangles are similar, their corresponding parts are related in specific ways based on the similarity ratio (or scale factor). If \( \triangle ABC \sim \triangle XYZ \) with a similarity ratio \( k \) (meaning \( \frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX} = k \)), then:

• Corresponding angles are equal: \( \angle A = \angle X \), \( \angle B = \angle Y \), \( \angle C = \angle Z \).
• The ratio of their perimeters is equal to the similarity ratio \( k \).
• The ratio of their corresponding altitudes, medians, and angle bisectors is equal to the similarity ratio \( k \).
• The ratio of their areas is equal to the square of the similarity ratio \( k^2 \).

In our problem, the similarity ratio from \( \triangle APQ \) to \( \triangle ABC \) is 0.4 (or 2/5). This means any linear measurement in the larger triangle ABC is \( 1/0.4 = 2.5 \) times the corresponding measurement in the smaller triangle APQ. Or, any linear measurement in the smaller triangle APQ is 0.4 (or 2/5) times the corresponding measurement in the larger triangle ABC.