Let's break down this geometry problem about triangle ABC. We are given some information about the triangle and asked to find the length of a specific perpendicular line segment, also known as an altitude.

Understanding Triangle Altitude and Area

An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side (or the extension of the opposite side). The length of the altitude is the height when the corresponding side is considered the base.

The area of a triangle can be calculated using the formula:

\(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)

In any triangle, the area remains constant regardless of which side is chosen as the base and which corresponding altitude is used as the height.

Using Given Information for Triangle ABC

We are given the following details for Δ ABC:

• Length of the perpendicular from vertex A to side BC is 15 cm. This is the altitude from A, let's call it \(h_a\). So, \(h_a = 15\) cm.
• Length of side BC is 8 cm. This is the base corresponding to \(h_a\), let's call it \(a\). So, \(a = 8\) cm.
• Length of side AC is 24 cm. This is another side, let's call it \(b\). So, \(b = 24\) cm.

We need to find the length of the perpendicular from vertex B to side AC. This is the altitude from B, let's call it \(h_b\).

Calculating Area of Δ ABC

We can calculate the area of Δ ABC using the given base BC (\(a\)) and its corresponding altitude \(h_a\):

\(\text{Area} = \frac{1}{2} \times \text{BC} \times h_a\)

\(\text{Area} = \frac{1}{2} \times 8 \text{ cm} \times 15 \text{ cm}\)

\(\text{Area} = \frac{1}{2} \times 120 \text{ cm}^2\)

\(\text{Area} = 60 \text{ cm}^2\)

Finding Perpendicular Length from B to AC

Now, we use the side AC (\(b\)) as the base and the unknown altitude from B (\(h_b\)) as the height to calculate the area of the same triangle:

\(\text{Area} = \frac{1}{2} \times \text{AC} \times h_b\)

We know the Area is 60 cm\(^2\) and AC is 24 cm. So, we can set up the equation:

\(60 \text{ cm}^2 = \frac{1}{2} \times 24 \text{ cm} \times h_b\)

Now, let's solve for \(h_b\):

\(60 = 12 \times h_b\)

\(h_b = \frac{60}{12}\)

\(h_b = 5 \text{ cm}\)

So, the length of the perpendicular from B to AC is 5 cm.

Summary of Calculation

The length of the perpendicular from B to AC is 5 cm.