Question

Pentagon models a running track in a park. a section of track connects point to point . triangle bcd with a rectangular base abde on the graph with points a (0, 6), b (2, 6), d (2, 0), e (0, 0), and c at (6, 5) two runners are taking separate routes through the park: one will follow the path along the perimeter of rectangle , and the other will follow the track along the perimeter of triangle . they decide to see who, if either, will run the farthest by finding the ratio of the perimeter of rectangle to the perimeter of triangle . what is the ratio of the perimeter of rectangle to the perimeter of triangle ?

Answer 1

To find the ratio of the perimeter of the rectangle to the perimeter of the triangle, calculate the lengths of the sides of both shapes and then divide the perimeter of the rectangle by the perimeter of the triangle.

Step-by-step explanation:

To find the ratio of the perimeter of the rectangle to the perimeter of the triangle, we need to calculate the lengths of the sides of both shapes.

The rectangle has side lengths AB = 2 units and AD = 6 units. So, the perimeter of the rectangle is 2(AB + AD) = 2(2 + 6) = 16 units.

The triangle has side lengths BC = 6 units, CD = 5 units, and DB = √((2 - 0)^2 + (0 - 5)^2) = √(4 + 25) = √29 units. So, the perimeter of the triangle is BC + CD + DB = 6 + 5 + √29.

To find the ratio, we can divide the perimeter of the rectangle by the perimeter of the triangle: 16 / (6 + 5 + √29).

Answer 2

The perimeter of rectangle ABDE is 16 units and the perimeter of triangle BCD is approximately 9.12 units. By calculating the sides using coordinates and applying the Pythagorean theorem, we find the ratio of the perimeter of the rectangle to the perimeter of the triangle is approximately 1.75.

To find the ratio of the perimeter of the rectangle ABDE to the perimeter of the triangle BCD, we first need to calculate the lengths of the sides of both shapes on the graph.

The rectangle is easier since it is a right-angled figure with horizontal and vertical sides.

The length AD (or BE) is the difference in the y-coordinates of points A and D (or B and E), which is 6 - 0 = 6 units.

The width AB (or DE) is the difference in the x-coordinates of points A and B (or D and E), which is 2 - 0 = 2 units.

Therefore, the perimeter of the rectangle is 2*(length + width) = 2*(6 + 2) = 16 units.

To find the length of sides BC and CD of the triangle, we will use the Pythagorean theorem since the coordinates given suggest right triangles between points B, C, and D internally.

BC is a vertical line, so its length is the difference in y-coordinates:

(5 - 6 = -1 unit, or just 1 unit considering length).

CD is a horizontal line, so its length is the difference in x-coordinates:

(6 - 2 = 4 units).

The hypotenuse BD can be found by applying the Pythagorean theorem (a² + b² = c²), where a and b are BC and CD we just calculated, and c is the hypotenuse BD. c² = 1² + 4² = 1 + 16 = 17, so BD (c) is the square root of 17, which is approximately 4.12 units.

The perimeter of the triangle is then the sum of its sides, which is:

BC + CD + BD = 1 + 4 + 4.12 ≈ 9.12 units.

The ratio of the perimeter of rectangle to the perimeter of triangle is 16 units / 9.12 units ≈ 1.75.

So, the runner following the perimeter of the rectangle will run farther compared to the runner following the perimeter of the triangle.