This regular octagon has a side length 15.0cm. Determine the distance from one vertex to the opposite vertex, measured through the centre of the octagon. Give your answer to to the nearest tenth of a centimeter.

Divide the octagon into 8 congruent isosceles triangles, each with base 15.

Since the vertex angle is 360/8 = 45°, each of the slant sides has length

7.5/sin22.5° = 19.6

The desired distance is twice the length of one of these sides, or 39.2

Where did you get 7.5 and 22...

To determine the distance from one vertex to the opposite vertex, measured through the center of the octagon, we need to find the length of the octagon's diagonal.

The diagonal of a regular octagon can be found by using the formula:
diagonal length = side length × √(2 + √2)

Given that the side length of the octagon is 15.0 cm, we can substitute this value into the formula:

diagonal length = 15.0 cm × √(2 + √2)

Calculating this value, we get:

diagonal length ≈ 15.0 cm × √(2 + √2) ≈ 15.0 cm × √(2 + 1.414) ≈ 15.0 cm × √(3.414)

diagonal length ≈ 15.0 cm × 1.845 = 27.675 cm

Therefore, the distance from one vertex to the opposite vertex, measured through the center of the octagon, is approximately 27.7 cm to the nearest tenth of a centimeter.

To determine the distance from one vertex to the opposite vertex of a regular octagon measured through the center, we need to find the length of the diagonal.

To find the diagonal of the octagon, we can use the properties of a regular octagon. Each diagonal of a regular octagon splits the octagon into two congruent right triangles.

Since we know the side length of the octagon (15.0 cm), we can use the Pythagorean theorem to find the length of the diagonal.

Let's denote the side length as "s" and the length of the diagonal as "d".

In a right triangle, the hypotenuse (diagonal) is related to the lengths of the other two sides by the Pythagorean theorem:

d^2 = s^2 + s^2

d^2 = 2s^2

Taking the square root of both sides to solve for "d", we get:

d = √(2s^2)

Plugging in the given side length of the octagon (15.0 cm), we have:

d = √(2 * (15.0 cm)^2)
= √(2 * 225.0 cm^2)
= √(450.0 cm^2)
≈ 21.2 cm

Therefore, the distance from one vertex to the opposite vertex of the regular octagon, measured through the center, is approximately 21.2 cm (to the nearest tenth of a centimeter).