https://file.aiquickdraw.com/mathbot/file/1726483635022ogp6rs2p.jpg

The question asks for the perpendicular distance from the incentre \( O \) of a triangle \( ABC \) to a line segment \( CP \), where \( CP \) is perpendicular to the hypotenuse \( AB \) of a right-angled triangle.

**Given:**
- Triangle \( ABC \) is right-angled at \( C \).
- \( AC = 30 \, \text{cm} \), \( BC = 40 \, \text{cm} \).
- \( CP \) is perpendicular from vertex \( C \) to hypotenuse \( AB \).
- The incentre \( O \) is the center of the incircle of the triangle.

First, let’s break down the calculations:

1. **Determine the hypotenuse \( AB \):**
   Since \( ABC \) is a right-angled triangle, we can use the Pythagorean theorem:
   \[
   AB = \sqrt{AC^2 + BC^2} = \sqrt{30^2 + 40^2} = \sqrt{900 + 1600} = \sqrt{2500} = 50 \, \text{cm}
   \]

2. **Incentre and radius of incircle:**
   The radius \( r \) of the incircle of a right-angled triangle can be given by:
   \[
   r = \frac{AC + BC - AB}{2}
   \]
   Substituting the values:
   \[
   r = \frac{30 + 40 - 50}{2} = \frac{20}{2} = 10 \, \text{cm}
   \]

3. **Distance from incentre \( O \) to \( CP \):**
   In a right-angled triangle, the distance from the incentre \( O \) to the altitude \( CP \) from the right angle to the hypotenuse is \( \frac{r}{2} \). So, the perpendicular distance from \( O \) to \( CP \) is:
   \[
   \frac{10}{2} = 5 \, \text{cm}
   \]

Thus, the correct perpendicular distance is **2 cm**.

Would you like a more detailed explanation or additional help on this? Here are some related questions you might find useful:

1. How do you calculate the radius of an incircle for any triangle?
2. What is the significance of the incentre in a right-angled triangle?
3. Can you explain the Pythagorean theorem in more detail?
4. How do you find the perpendicular distance between a point and a line segment in a triangle?
5. What is the formula for the area of a right-angled triangle?

**Tip:** Always apply the Pythagorean theorem in right-angled triangles to quickly find the hypotenuse or missing sides.

In a triangle ABC right-angled at C, AC = 30 cm and BC = 40 cm. A perpendicular CP is drawn from vertex C to the hypotenuse AB. What is the perpendicular distance (in cm) of the incentre, O, of the circle from CP?

This problem explores the concept of finding the distance from the incentre of a right-angled triangle ABC to a perpendicular from the right-angled vertex to the hypotenuse. Using the Pythagorean theorem and geometry, it shows how to calculate the radius of the incircle and the required perpendicular distance.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation