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The coordinates for a rhombus are given as (2a, 0), (0, 2b), (–2a, 0), and (0, –2b). Write a plan to prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry. Be sure to include the formulas.
A rhombus is drawn on a coordinate plane and the midpoints of all 4 sides are plotted.The upper and lower vertices of the rhombus are located on the y-axis.
the left and right vertices are located on the x-axis.
The upper vertex has coordinates left parentheses 0 comma 2 b right parentheses.
The midpoint of the upper right side has coordinates left parentheses a comma b right parentheses.
The right vertex has coordinates left parentheses 2 a comma 0 right parentheses.
The midpoint of the lower right side has coordinates left parentheses a comma negative b right parentheses.
The lower vertex has coordinates left parentheses 0 comma negative 2 b right parentheses.
The midpoint of the lower left side has coordinates left parentheses negative a comma negative b right parentheses.
The left vertex has coordinates left parentheses negative 2 a comma 0 right parentheses.
The midpoint of the upper left side has coordinates left parentheses negative a comma b right parentheses.
(3 points)
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To prove that the midpoints of the sides of a rhombus determine a rectangle using coordinate geometry, we can follow the following steps:
1. Define the coordinates for the midpoints of the sides of the rhombus:
- Midpoint of the upper side: (a, b)
- Midpoint of the right side: (2a, 0)
- Midpoint of the lower side: (a, -b)
- Midpoint of the left side: (-2a, 0)
2. Calculate the slopes of opposite sides of the rhombus:
- Slope of the upper side: m1 = (b - 0)/(a - 0) = b/a
- Slope of the lower side: m2 = (-b - 0)/(a - 0) = -b/a
- Slope of the right side: m3 = (0 - b)/(2a - a) = -b/a
- Slope of the left side: m4 = (0 - (-b))/(-2a - a) = b/a
3. Since the opposite sides of a rhombus have equal slopes, we can compare the slopes obtained:
- Slope of the upper side (m1) = Slope of the lower side (m2)
- Slope of the right side (m3) = Slope of the left side (m4)
4. Calculate the product of the slopes:
- Product of the slopes of the opposite sides = (b/a) * (-b/a) = -1
5. Since the product of the slopes of opposite sides is -1, it indicates that the sides are perpendicular to each other.
6. Therefore, based on the properties of a rectangle, the midpoints of the sides of the rhombus determine a rectangle.
Bot GPT 3.5 please shorten
To prove that the midpoints of a rhombus determine a rectangle using coordinate geometry:
1. Define the coordinates for the midpoints of the sides.
2. Calculate the slopes of opposite sides.
3. Compare the slopes to check for perpendicularity.
4. Calculate the product of the slopes.
5. If the product is -1, the sides are perpendicular.
6. Thus, the midpoints determine a rectangle.