Question

The diagonals of a parallelogram bisect each other. Steps (a), (b), and (c) outline a proof of this theorem. (See Exercise 25 for a particular instance of this theorem.) (a) In the parallelogram OABC shown in the figure, check that the coordinates of $B$ must be $(a+b, c)$ (b) Use the midpoint formula to calculate the midpoints of diagonals $\overline{O B}$ and $\overline{A C}$ (c) The two answers in part (b) are identical. This shows that the two diagonals do indeed bisect each other, as we wished to prove. (FIGURE CAN'T COPY)

Name: Problem 29, Chapter 1: Precalculus
Uploaded: 2020-07-27T19:48:35-08:00
Duration: 3 min 2 s
Channel: Charles Carter

   The diagonals of a parallelogram bisect each other. Steps
(a), (b), and (c) outline a proof of this theorem. (See Exercise 25 for a particular instance of this theorem.)
(a) In the parallelogram OABC shown in the figure, check that the coordinates of $B$ must be $(a+b, c)$
(b) Use the midpoint formula to calculate the midpoints of diagonals $\overline{O B}$ and $\overline{A C}$
(c) The two answers in part (b) are identical. This shows that the two diagonals do indeed bisect each other, as we wished to prove.
(FIGURE CAN'T COPY)

Precalculus

David Cohen,… 4th Edition

Chapter 1, Problem 29

Instant Answer

Solved by Expert Charles Carter

07/27/2020

Step 1

Since B is the opposite vertex of O and parallelogram's opposite sides are parallel and equal in length, the x-coordinate of B will be the sum of x-coordinates of A and C, and the y-coordinate of B will be the same as that of C. Hence, the coordinates of B will be Show more…

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The diagonals of a parallelogram bisect each other. Steps (a), (b), and (c) outline a proof of this theorem. (See Exercise 25 for a particular instance of this theorem.) (a) In the parallelogram OABC shown in the figure, check that the coordinates of $B$ must be $(a+b, c)$ (b) Use the midpoint formula to calculate the midpoints of diagonals $\overline{O B}$ and $\overline{A C}$ (c) The two answers in part (b) are identical. This shows that the two diagonals do indeed bisect each other, as we wished to prove. (FIGURE CAN'T COPY)

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Key Concepts

Properties of Parallelograms

This concept refers to the defining attributes of parallelograms, such as having opposite sides that are equal and parallel. One of the important properties is that the diagonals of a parallelogram bisect each other, which means each diagonal cuts the other into two equal parts.

Midpoint Formula

The midpoint formula is a fundamental tool in coordinate geometry, used to calculate the point that divides a line segment into two equal parts. It is defined as ((x1 + x2) / 2, (y1 + y2) / 2) for any segment with endpoints (x1, y1) and (x2, y2), and is used to show that both diagonals of a parallelogram share the same midpoint.

Coordinate Geometry

Coordinate geometry, or analytic geometry, involves representing geometric figures using coordinates and algebra. It allows geometric properties, such as those in a parallelogram, to be demonstrated through algebraic manipulation and calculation, providing a clear and rigorous method of proof.

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