Danny has a yard in the shape of a parallelogram:
width- (50 m)
length- (60 m)
height- (60 m)
he want to split it into 3 pieces, with more than one type of shape. they should all have equal perimeter. the perimeter can not be in decimals.
(Help please)
To split the yard into 3 pieces with equal perimeters, you can divide it into two rectangular sections and one triangular section.
1. Start by dividing the yard into two equal rectangles by drawing a straight line through the yard's center parallel to the width. This will create two rectangles, each with a width of 25 meters and a length of 60 meters.
2. Now, to create the third piece with a triangular shape, we need to calculate the perimeter of each rectangle. The perimeter of a rectangle is calculated by adding up the lengths of all four sides.
- For each rectangle, the perimeter is: 2(width + length) = 2(25 + 60) = 2(85) = 170 meters.
3. To create a triangular section with an equal perimeter, we need to determine the base and height of the triangle. Since the triangle's perimeter must be a whole number and the dimensions of the yard are given in whole numbers, it is likely that the base and height of the triangle will also be whole numbers.
4. Let's assume the base of the triangle is "b" meters. To determine the height, we can use the fact that the sum of all three sides of the triangle is equal to the perimeter of the rectangles (170 meters).
- The perimeter of a triangle is calculated by adding up the lengths of all three sides.
- Since we have two sides (the base and the height), we need to find the third side to calculate the perimeter.
- The third side can be found using the Pythagorean theorem: hypotenuse^2 = base^2 + height^2.
- Using this equation, we have hypotenuse^2 = b^2 + h^2.
5. To find a whole number solution, we can test different values of the base and calculate the corresponding height of the triangle using the Pythagorean theorem.
- For example, let's assume the base is 10 meters. Then we have, hypotenuse^2 = 10^2 + h^2.
- To obtain a whole number for the height, the hypotenuse^2 must be a perfect square. We can check different values for the hypotenuse squared until we find a perfect square.
- Let's start with hypotenuse^2 = 170 - 10^2 = 100.
- The square root of 100 is 10, which means the hypotenuse is 10 meters.
- Now we can use the Pythagorean theorem to find the height: 10^2 = 10^2 + h^2. Solving for h, we have h = 0.
6. In this case, the height is 0, which means the triangular section does not exist. This is because the given dimensions do not allow for an equal perimeter split into three pieces with different shapes.
Therefore, based on the given dimensions (50 m width, 60 m length, and 60 m height), it is not possible to split the yard into three pieces with equal perimeters and more than one type of shape.