Question

A polygon that is both equilateral and equiangular is called regular. Prove that all diagonals of a regular pentagon have the same length.

Answer 1

A regular pentagon is both equilateral and equiangular. We can prove that all diagonals of a regular pentagon have the same length by using the properties of triangles and the fact that the sum of the interior angles of a polygon with n sides is given by (n-2) × 180 degrees.

Step-by-step explanation:

A regular polygon is both equilateral and equiangular, which means that all its sides are equal in length and all its angles are equal in measure. In a regular pentagon, all five sides are equal and all five angles are equal. To prove that all diagonals of a regular pentagon have the same length, we can use the properties of triangles and the fact that the sum of the interior angles of a polygon with n sides is given by (n-2) × 180 degrees.

1. Let's consider a regular pentagon ABCDE, with sides of length s.
2. Draw the diagonals connecting each vertex of the pentagon to every other vertex. There are five diagonals: AC, AD, AE, BD, and BE.
3. Observe that each diagonal forms a triangle with two sides equal in length, which are the sides of the regular pentagon. Let's take triangle ABC as an example.
4. Since triangle ABC has two sides of length s, it is an isosceles triangle. In an isosceles triangle, the base angles are equal. Therefore, angle BAC is equal to angle BCA.
5. Using the fact that the sum of the angles in a triangle is 180 degrees, we can deduce that angle BAC + angle ABC + angle BCA = 180 degrees.
6. Substituting angle BAC with angle BCA (since they are equal), we get angle ABC + angle ABC + angle ABC = 180 degrees.
7. Simplifying the equation, we have 3 × angle ABC = 180 degrees.
8. Dividing both sides of the equation by 3, we find that angle ABC = 60 degrees.
9. Since angle ABC is equal to angle BCA and angle BCA is equal to angle CAB, all angles in the regular pentagon ABCDE are equal to 60 degrees.
10. Now, let's consider the diagonals. For example, let's look at diagonal AC.
11. Triangle ABC has angles of 60 degrees, 60 degrees, and 60 degrees.
12. Since the sum of the angles in a triangle is 180 degrees, we can deduce that the third angle in triangle ABC is also 60 degrees.
13. Therefore, triangle ABC is an equilateral triangle, which means that all sides of the triangle are equal in length (s in this case).
14. Since every diagonal of the regular pentagon forms an equilateral triangle with two sides equal to s, we can conclude that all diagonals of the regular pentagon have the same length as the sides of the pentagon.

Answer 2

The diagonals of a regular pentagon are congruent because they are all sides of congruent isosceles triangles formed within the pentagon, derived from its equiangular and equilateral properties, and supported by the transitive property of equality.

Step-by-step explanation:

In proving that all diagonals of a regular pentagon have the same length, we look to the properties that define it: a polygon that is both equilateral and equiangular.

To prove the diagonals are congruent, consider any two adjacent vertices of the pentagon and label them as A and B. Since the pentagon is equilateral, we know AB is equal in length to the other sides. Now, consider the vertex opposite to side AB and label it as C. The diagonal AC is drawn from A to C. Since the pentagon is also equiangular, angles at each vertex are equal, forming isosceles triangles within the pentagon when diagonals are drawn.

By drawing all possible diagonals from vertex A, we form a series of triangles ACD, ACE, and so on. The Pythagorean theorem suggests that the sides of these triangles relate such that the square of one side plus the square of another side equals the square of the hypotenuse. Nonetheless, in a regular pentagon, the sides and angles are equal, so all isosceles triangles formed are congruent, and thus, all diagonals are equal in length.

This conclusion relies on the transitive property of equality (if AC = AE and AE = AD, then AC = AD), reaffirming that all diagonals (AC, AD, AE, etc.) are congruent. The constant properties of the regular pentagon allow us to determine that all the diagonals share the same length without individually measuring each one.