https://file.aiquickdraw.com/mathbot/file/1725975133541nwd5svco.jpg

In the given figure:

- AB is the diameter of the semicircle, and the lengths of segments PA and PB are provided:
  - PA = 6 cm
  - PB = 4 cm

We are asked to find:
1. The length of PC.
2. Draw an equilateral triangle with PC as a side.

### Step 1: Find the length of PC
Using the Pythagorean theorem, we can calculate PC. Point P, where the perpendicular meets AB, forms a right triangle with points A and B, and C lies on the semicircle's circumference.

Since AB is the diameter, and P lies on the diameter:
- \( PA = 6 \, \text{cm} \)
- \( PB = 4 \, \text{cm} \)

The total length of AB = \( PA + PB = 6 + 4 = 10 \, \text{cm} \), so the radius of the semicircle is 5 cm.

To find PC, we use the following relationships in a right triangle formed by the perpendicular:
- PA = 6
- PB = 4

Applying the Pythagorean theorem to the triangle:
\[
PC = \sqrt{PA^2 + PB^2}
\]
Let's calculate:

\[
PC = \sqrt{6^2 + 4^2} = \sqrt{36 + 16} = \sqrt{52} = \sqrt{4 \times 13} = 2\sqrt{13} \, \text{cm}
\]

### Step 2: Drawing an Equilateral Triangle
To draw an equilateral triangle with PC as one of its sides, follow these steps:
1. Use \( PC = 2\sqrt{13} \, \text{cm} \) as the base of the triangle.
2. Construct the two other sides equal to this length.
3. Ensure the angles between all sides are \( 60^\circ \), which is characteristic of an equilateral triangle.

Would you like to discuss the details or need more help with the drawing part?

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Here are some related questions to deepen your understanding:
1. How do you use the Pythagorean theorem in right triangles?
2. Can the same method be applied to find other distances in similar geometrical figures?
3. What is the significance of the diameter of a semicircle in problems like this?
4. How does knowing the radius of the circle help in solving geometry problems?
5. What are some alternative methods for constructing an equilateral triangle?

**Tip:** Always check the geometry of the figure and ensure the properties of the shapes (like right angles, perpendiculars, or diameters) are clear before solving problems.

In the figure, AB is the diameter of the semicircle. PA = 6 cm, PB = 4 cm. Find PC and draw an equilateral triangle with PC as a side.

Learn how to find the length of PC in a semicircle where AB is the diameter and PA and PB are known. This problem involves using the Pythagorean Theorem and constructing an equilateral triangle, suitable for Grades 7-9 students.

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