Answer :
To solve this problem, let's transform it step-by-step:
1. Draw the Pentagon:
- A regular pentagon has 5 equal sides. Here, each side is given as 30mm.
- A regular pentagon can be inscribed within a circle, so let's first determine the vertices of the pentagon on this circle by calculating their coordinates. Note that the given vertices are based on the center being at the origin (0,0) coordinate.
- The vertices of the pentagon can be calculated by using polar coordinates and converting them to Cartesian coordinates. The angles in this case are multiples of [tex]\( \frac{2\pi}{5} \)[/tex] (i.e., [tex]\( \frac{360^\circ}{5} = 72^\circ \)[/tex]).
The vertices of the pentagon are:
```
(30.0, 0.0)
(9.270509831248424, 28.531695488854606)
(-24.27050983124842, 17.633557568774197)
(-24.270509831248425, -17.63355756877419)
(9.270509831248416, -28.53169548885461)
```
By plotting these points and connecting them in sequence, you will get a regular pentagon.
2. Construct the Circle:
- Draw a circle of radius 60mm. This circle will serve as the boundary for inscribing the square.
3. Inscribe a Square in the Circle:
- A square inscribed in a circle of radius [tex]\(r\)[/tex] will have its vertices on the circle.
- The diagonal of the square will be equal to the diameter of the circle. So if the radius of the circle is 60mm, the diagonal of the square will be [tex]\(2 \times 60 = 120 \text{mm}\)[/tex].
- The side length [tex]\(s\)[/tex] of the square can be found using the Pythagorean theorem in one of the triangles formed by cutting the square along its diagonal. This gives [tex]\( s = \text{diagonal} / \sqrt{2} \)[/tex].
The side length of the square, therefore, is:
```
120 / sqrt(2) ≈ 84.8528137423857 mm
```
- To find the vertices, we center the square at the origin (0,0) and calculate its vertices, which will be equidistant from the origin along the axes of symmetry of the square.
The vertices of the square are:
```
(-42.42640687119285, -42.42640687119285)
(42.42640687119285, -42.42640687119285)
(42.42640687119285, 42.42640687119285)
(-42.42640687119285, 42.42640687119285)
```
By plotting these points and connecting them in sequence, you will get a square inscribed within the circle.
In summary:
- You have your pentagon vertices: (30.0, 0.0), (9.270509831248424, 28.531695488854606), (-24.27050983124842, 17.633557568774197), (-24.270509831248425, -17.63355756877419), and (9.270509831248416, -28.53169548885461).
- You have your square vertices: (-42.42640687119285, -42.42640687119285), (42.42640687119285, -42.42640687119285), (42.42640687119285, 42.42640687119285), and (-42.42640687119285, 42.42640687119285).
1. Draw the Pentagon:
- A regular pentagon has 5 equal sides. Here, each side is given as 30mm.
- A regular pentagon can be inscribed within a circle, so let's first determine the vertices of the pentagon on this circle by calculating their coordinates. Note that the given vertices are based on the center being at the origin (0,0) coordinate.
- The vertices of the pentagon can be calculated by using polar coordinates and converting them to Cartesian coordinates. The angles in this case are multiples of [tex]\( \frac{2\pi}{5} \)[/tex] (i.e., [tex]\( \frac{360^\circ}{5} = 72^\circ \)[/tex]).
The vertices of the pentagon are:
```
(30.0, 0.0)
(9.270509831248424, 28.531695488854606)
(-24.27050983124842, 17.633557568774197)
(-24.270509831248425, -17.63355756877419)
(9.270509831248416, -28.53169548885461)
```
By plotting these points and connecting them in sequence, you will get a regular pentagon.
2. Construct the Circle:
- Draw a circle of radius 60mm. This circle will serve as the boundary for inscribing the square.
3. Inscribe a Square in the Circle:
- A square inscribed in a circle of radius [tex]\(r\)[/tex] will have its vertices on the circle.
- The diagonal of the square will be equal to the diameter of the circle. So if the radius of the circle is 60mm, the diagonal of the square will be [tex]\(2 \times 60 = 120 \text{mm}\)[/tex].
- The side length [tex]\(s\)[/tex] of the square can be found using the Pythagorean theorem in one of the triangles formed by cutting the square along its diagonal. This gives [tex]\( s = \text{diagonal} / \sqrt{2} \)[/tex].
The side length of the square, therefore, is:
```
120 / sqrt(2) ≈ 84.8528137423857 mm
```
- To find the vertices, we center the square at the origin (0,0) and calculate its vertices, which will be equidistant from the origin along the axes of symmetry of the square.
The vertices of the square are:
```
(-42.42640687119285, -42.42640687119285)
(42.42640687119285, -42.42640687119285)
(42.42640687119285, 42.42640687119285)
(-42.42640687119285, 42.42640687119285)
```
By plotting these points and connecting them in sequence, you will get a square inscribed within the circle.
In summary:
- You have your pentagon vertices: (30.0, 0.0), (9.270509831248424, 28.531695488854606), (-24.27050983124842, 17.633557568774197), (-24.270509831248425, -17.63355756877419), and (9.270509831248416, -28.53169548885461).
- You have your square vertices: (-42.42640687119285, -42.42640687119285), (42.42640687119285, -42.42640687119285), (42.42640687119285, 42.42640687119285), and (-42.42640687119285, 42.42640687119285).
