Posted: 1/29/2011 1:22:05 PM EDT
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My homework assignment is to come up with mathematical proofs for the theorems in the book. Here is the theorem: The segments that join the midpoints of the consecutive sides of a plane quadrilateral form a parallelogram. So I graphed a plane quadrilateral, calculated the midpoints and graphed the segments connecting the midpoints of the consecutive sides of the quadrilateral. Now I am writing a conditional statement that will define how I will determine if the line segments connecting the midpoints of the consecutive sides of the quadrilateral form a parallelogram. Basically, I am telling the professor what I intend to calculate to determine if the shape is a parallelogram or not, and what the expected results will be if the shape is a parallelogram. What I want to know is if this statement (my statement) makes sense: "If the length and slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are equal to one another, then the shape formed by the line segments connecting the midpoints of the consecutive sides of the quadrilateral is a parallelogram" Is this true and does it make sense? |
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The definition of a parallelogram is a quadrilateral with opposing sides parrelel. It is sufficient to prove the quadrilateral is a parallelogram by demonstrating that each pair of opposite sides are parallel.
ETA: I guess you would also have to show that there are four sides. 
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Quoted: The definition of a parallelogram is a quadrilateral with opposing sides parrelel. It is sufficient to prove the quadrilateral is a parallelogram by demonstrating that each pair of opposite sides are parallel. ETA: I guess you would also have to show that there are four sides. So then: "If the slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are equal to one another then the shape formed by the line segments connecting the midpoints of the opposite sides of the quadrilateral is a parallelogram"? |
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The definition of a parallelogram is a quadrilateral with opposing sides parrelel. It is sufficient to prove the quadrilateral is a parallelogram by demonstrating that each pair of opposite sides are parallel. ETA: I guess you would also have to show that there are four sides. So then: "If the slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are equal to one another then the shape formed by the line segments connecting the midpoints of the opposite sides of the quadrilateral is a parallelogram"? Yep I think also that if you simply show that the opposite sides are the same length then you have a parallelogram since then the slopes have to be equal. |
| By "consecutive sides" I'm assuming you mean the two lines of the quadrilateral that make an "L" shape. So you're saying, 'The line segments that connect the midpoints of the lines of the two L shapes of the quadrilateral are parallel if they have the same length and slope'. Makes sense. |
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Quoted: My homework assignment is to come up with mathematical proofs for the theorems in the book. Here is the theorem: The segments that join the midpoints of the consecutive sides of a plane quadrilateral form a parallelogram. So I graphed a plane quadrilateral, calculated the midpoints and graphed the segments connecting the midpoints of the consecutive sides of the quadrilateral. Now I am writing a conditional statement that will define how I will determine if the line segments connecting the midpoints of the consecutive sides of the quadrilateral form a parallelogram. Basically, I am telling the professor what I intend to calculate to determine if the shape is a parallelogram or not, and what the expected results will be if the shape is a parallelogram. What I want to know is if this statement (my statement) makes sense: "If the length and slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are equal to one another, then the shape formed by the line segments connecting the midpoints of the consecutive sides of the quadrilateral is a parallelogram" Is this true and does it make sense? What you are saying is not a mathematical proof. It's simply a statement. You need to use the lines connecting the midpoints through the centroid of the quadrilateral. You can show that the angles and sides of the lines going through the centroid are the same and therefore lengths and angles of the lines connecting the midpoints are the same length. And that the angles are congruent. The proof needs to be step by step facts such as "vertical angles formed by intersecting lines are congruent." |
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Quoted:
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The definition of a parallelogram is a quadrilateral with opposing sides parrelel. It is sufficient to prove the quadrilateral is a parallelogram by demonstrating that each pair of opposite sides are parallel. ETA: I guess you would also have to show that there are four sides. So then: "If the slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are equal to one another then the shape formed by the line segments connecting the midpoints of the opposite sides of the quadrilateral is a parallelogram"? Yes. |
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Quoted: Quoted: My homework assignment is to come up with mathematical proofs for the theorems in the book. Here is the theorem: The segments that join the midpoints of the consecutive sides of a plane quadrilateral form a parallelogram. So I graphed a plane quadrilateral, calculated the midpoints and graphed the segments connecting the midpoints of the consecutive sides of the quadrilateral. Now I am writing a conditional statement that will define how I will determine if the line segments connecting the midpoints of the consecutive sides of the quadrilateral form a parallelogram. Basically, I am telling the professor what I intend to calculate to determine if the shape is a parallelogram or not, and what the expected results will be if the shape is a parallelogram. What I want to know is if this statement (my statement) makes sense: "If the length and slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are equal to one another, then the shape formed by the line segments connecting the midpoints of the consecutive sides of the quadrilateral is a parallelogram" Is this true and does it make sense? What you are saying is not a mathematical proof. It's simply a statement. You need to use the lines connecting the midpoints through the centroid of the quadrilateral. You can show that the angles and sides of the lines going through the centroid are the same and therefore lengths and angles of the lines connecting the midpoints are the same length. And that the angles are congruent. The proof needs to be step by step facts such as "vertical angles formed by intersecting lines are congruent." I understand that. The key phrase in my post is: "Basically, I am telling the professor what I intend to calculate to determine if the shape is a parallelogram or not, and what the expected results will be if the shape is a parallelogram." So, I am stating what I will be calculating and what I expect the outcome of those calculations to be in order to prove the theorem correct. For this problem, I went on to calculate the midpoints of the line segments forming the quadrilateral and then calculated the slope of the lines connecting the midpoints of the consecutive sides of the quadrilateral. By showing that the slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are the same, I proved that the shape formed is a parallelogram. I did the calculations and showed them. All my statement does is explain the math I am using and what it means to the theorem. Since math proofs can be approached in different ways, I felt that this statement was necessary in order to show the professor that I know what I'm talking about and so that he can understand how I tackled it. |
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My homework assignment is to come up with mathematical proofs for the theorems in the book. Here is the theorem: The segments that join the midpoints of the consecutive sides of a plane quadrilateral form a parallelogram. So I graphed a plane quadrilateral, calculated the midpoints and graphed the segments connecting the midpoints of the consecutive sides of the quadrilateral. Now I am writing a conditional statement that will define how I will determine if the line segments connecting the midpoints of the consecutive sides of the quadrilateral form a parallelogram. Basically, I am telling the professor what I intend to calculate to determine if the shape is a parallelogram or not, and what the expected results will be if the shape is a parallelogram. What I want to know is if this statement (my statement) makes sense: "If the length and slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are equal to one another, then the shape formed by the line segments connecting the midpoints of the consecutive sides of the quadrilateral is a parallelogram" Is this true and does it make sense? What you are saying is not a mathematical proof. It's simply a statement. You need to use the lines connecting the midpoints through the centroid of the quadrilateral. You can show that the angles and sides of the lines going through the centroid are the same and therefore lengths and angles of the lines connecting the midpoints are the same length. And that the angles are congruent. The proof needs to be step by step facts such as "vertical angles formed by intersecting lines are congruent." That is one way to do it; calculating the slopes for all four lines using generalized points (a,b; c,d; etc) and showing opposite sides are parallel is another way to do it. The OP specifically states that this is his explanation of his calculation, not his actual proof. |
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Quoted: snip I understand that. The key phrase in my post is: "Basically, I am telling the professor what I intend to calculate to determine if the shape is a parallelogram or not, and what the expected results will be if the shape is a parallelogram." So, I am stating what I will be calculating and what I expect the outcome of those calculations to be in order to prove the theorem correct. For this problem, I went on to calculate the midpoints of the line segments forming the quadrilateral and then calculated the slope of the lines connecting the midpoints of the consecutive sides of the quadrilateral. By showing that the slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are the same, I proved that the shape formed is a parallelogram. I did the calculations and showed them. All my statement does is explain the math I am using and what it means to the theorem. Since math proofs can be approached in different ways, I felt that this statement was necessary in order to show the professor that I know what I'm talking about and so that he can understand how I tackled it. Showing the calculations is not a mathematical proof either. You want to provide a step by step proof using geometry relationships that will show that it is true for any quadrilateral, not just the one given. |
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Quoted: Quoted: snip I understand that. The key phrase in my post is: "Basically, I am telling the professor what I intend to calculate to determine if the shape is a parallelogram or not, and what the expected results will be if the shape is a parallelogram." So, I am stating what I will be calculating and what I expect the outcome of those calculations to be in order to prove the theorem correct. For this problem, I went on to calculate the midpoints of the line segments forming the quadrilateral and then calculated the slope of the lines connecting the midpoints of the consecutive sides of the quadrilateral. By showing that the slope of the opposite line segments connecting the midpoints of the consecutive sides of the quadrilateral are the same, I proved that the shape formed is a parallelogram. I did the calculations and showed them. All my statement does is explain the math I am using and what it means to the theorem. Since math proofs can be approached in different ways, I felt that this statement was necessary in order to show the professor that I know what I'm talking about and so that he can understand how I tackled it. Showing the calculations is not a mathematical proof either. You want to provide a step by step proof using geometry relationships that will show that it is true for any quadrilateral, not just the one given. Yes, I understand this also. I used generic points for X and Y and then calculated the formulas for the slope of each line using those generic points to show that the slopes are equal to one another. This is exactly how they do it in the book. |
