https://file.aiquickdraw.com/mathbot/file/1727495012181zbt770q5.jpg

Let's break down and solve the given problem based on the diagram and description provided.

### Diagram Overview:
The diagram shows an equilateral triangle \( PQR \), with side length 5 cm. The points are defined as follows:
- \( M \) is the midpoint of line \( QR \).
- There is an arc of a circle with center at \( P \) that touches \( QR \) at \( M \) and meets \( PQ \) at \( X \) and \( PR \) at \( Y \).
- We are tasked with finding two things in terms of \( \pi \) and \( \sqrt{3} \):
  1. The total perimeter of the shaded region.
  2. The total area of the shaded region.

### Part (a) - Perimeter of the shaded region
- The perimeter of the shaded region consists of three segments:
  1. The line segment \( MX \) along \( PQ \).
  2. The line segment \( MY \) along \( PR \).
  3. The arc from \( X \) to \( Y \), which is part of the circle centered at \( P \).

#### Step 1: Line Segments \( MX \) and \( MY \)
Since \( PQR \) is an equilateral triangle, each side has a length of 5 cm. The altitude of an equilateral triangle can be calculated as:
\[
h = \frac{\sqrt{3}}{2} \times 5 = \frac{5\sqrt{3}}{2} \, \text{cm}
\]
The midpoint \( M \) divides the base \( QR \) into two equal parts of 2.5 cm each. Because of symmetry, the lengths of \( MX \) and \( MY \) will each be half the altitude, giving:
\[
MX = MY = \frac{5\sqrt{3}}{2}
\]

#### Step 2: Arc Length
The arc is part of a circle centered at \( P \), and the angle subtended by the arc at \( P \) is \( 60^\circ \), or \( \frac{\pi}{3} \) radians. The radius of the circle is the side length of the equilateral triangle, which is 5 cm. The length of an arc is given by:
\[
\text{Arc Length} = r \times \theta = 5 \times \frac{\pi}{3} = \frac{5\pi}{3}
\]

#### Total Perimeter
The total perimeter of the shaded region is the sum of the two line segments and the arc:
\[
\text{Perimeter} = MX + MY + \text{Arc Length} = \frac{5\sqrt{3}}{2} + \frac{5\sqrt{3}}{2} + \frac{5\pi}{3} = 5\sqrt{3} + \frac{5\pi}{3}
\]

### Part (b) - Area of the shaded region
The area of the shaded region consists of the sector of the circle minus the area of triangle \( PXY \).

#### Step 1: Area of the Sector
The area of a sector of a circle is given by:
\[
\text{Area of Sector} = \frac{1}{2} r^2 \theta = \frac{1}{2} \times 5^2 \times \frac{\pi}{3} = \frac{25\pi}{6}
\]

#### Step 2: Area of Triangle \( PXY \)
The area of triangle \( PQR \) is:
\[
\text{Area of } PQR = \frac{1}{2} \times 5 \times \frac{5\sqrt{3}}{2} = \frac{25\sqrt{3}}{4}
\]
Since \( PXY \) is half of \( PQR \) (because \( M \) is the midpoint and the arc splits the triangle into two equal parts), the area of triangle \( PXY \) is:
\[
\text{Area of } PXY = \frac{1}{2} \times \frac{25\sqrt{3}}{4} = \frac{25\sqrt{3}}{8}
\]

#### Step 3: Total Shaded Area
The total shaded area is the area of the sector minus the area of triangle \( PXY \):
\[
\text{Shaded Area} = \frac{25\pi}{6} - \frac{25\sqrt{3}}{8}
\]

### Final Answers:
- **Perimeter**: \( 5\sqrt{3} + \frac{5\pi}{3} \)
- **Area**: \( \frac{25\pi}{6} - \frac{25\sqrt{3}}{8} \)

Let me know if you would like more details or have any questions.

#### Related Questions:
1. What is the total length of the sides of the equilateral triangle?
2. How do you calculate the height of an equilateral triangle given the side length?
3. What is the general formula for the length of an arc of a circle?
4. How do you find the area of a sector given the radius and angle?
5. Can you explain how symmetry helps simplify problems involving equilateral triangles?

**Tip:** In geometry problems, identifying symmetry and applying basic formulas for areas and perimeters can help break down complex shapes.

The diagram shows an equilateral triangle, PQR, with side length 5 cm. M is the midpoint of the line QR. An arc of a circle, centre P, touches QR at M and meets PQ at X and PR at Y. Find in terms of π and √3: (a) the total perimeter of the shaded region (b) the total area of the shaded region.

Learn how to find the total perimeter and area of a shaded region in an equilateral triangle with a circle arc. This problem involves geometry, including calculating arc length and sector areas, and applying properties of equilateral triangles. Solutions are provided in terms of π and √3.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation