Question

Julia is writing a coordinate proof to show that the diagonals of a parallelogram bisect each other. She starts by assigning coordinates as given.
Drag and drop the correct answer into each box to complete the proof.

The coordinates of point C are (__, c).

The coordinates of the midpoint of diagonal AC¯¯¯¯¯ are (__, c/2 ).

The coordinates of the midpoint of diagonal BD¯¯¯¯¯ are ( a+b/2, __).

AC¯¯¯¯¯ and BD¯¯¯¯¯ intersect at point E with coordinates ​ (a+b/2, c/2) ​ .

By the definition of midpoint, AE¯¯¯¯¯≅ __ and BE¯¯¯¯¯≅ __.

Therefore, diagonals AC¯¯¯¯¯ and BD¯¯¯¯¯ bisect each other.

Options:

1. a + b
2. a + c
3. b + c
4. a+b/2
5. a−b/2
6. a/2
7. b/2
8. c/2
9. AC¯¯¯¯¯
10. BD¯¯¯¯¯
11. CE¯¯¯¯¯
12. DE¯¯¯¯¯

Answer 1

Answer:

1. $C= (a+b, c)$ 1 2. $E=(\frac{a+b}{2},\frac{c}{2})$ 4 3. $\frac{c}{2}$ 8 4. AC,  BD 5. CE, DE 6.

Step-by-step explanation:

Below each answer the respective explanation, and also a numerical coordinate parallelogram just to make it clearer.

According to the graph

A= (0,0)

B= (a,0)

C= (a+b, c)

D=(b,c)

E=(a+b/2, c/2)

1. The coordinates of point C are $\overline{AB}=\overline{CD} = \Rightarrow (0-a) \Rightarrow a$

To find the x-coordinate of C: just add a to b

$C= (a+b, c)$

2. The coordinates of the midpoint of diagonal AC are (__, c/2 ).

Midpoints are calculated by:

$M=\left ( \frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} \right )$

Since A (0,0) and C (a+b,c) Then: (a+b/2, c/2)

$E=(\frac{a+b}{2},\frac{c}{2})$

3. The coordinates of the midpoint of diagonal BD are (a+b/2, __).

Then $\frac{c}{2}$ (8)

4. ___ and ___ intersect at point E with coordinates ​ (a+b/2, c/2)

AC and BD . True. This is a consequence of steps 2 and 3.

5. By the definition of midpoint, AE≅ __ and BE≅ __.

$\overline{AE} \cong\overline{CE}\: and\: \overline{BE}\cong \overline{DE}$

6.Therefore, diagonals AC and BD bisect each other.

In this sense, to bisect is to equally divide into two congruent line segments.

Answer 2

Answer:

Step-by-step explanation:

The coordinates of point C are (a+b__, c).

The coordinates of the midpoint of diagonal AC¯¯¯¯¯ are ((a+b)/2__, c/2 ).

The coordinates of the midpoint of diagonal BD¯¯¯¯¯ are ( a+b/2, _c/2_).

AC¯¯¯¯¯ and BD¯¯¯¯¯ intersect at point E with coordinates ​ (a+b/2, c/2) ​ .

By the definition of midpoint, AE¯¯¯¯¯≅ CE__ and BE¯¯¯¯¯≅ _DE_.

Therefore, diagonals AC¯¯¯¯¯ and BD¯¯¯¯¯ bisect each other.  

Options:  

1. a + b  

2. a + c  

3. b + c  

4. a+b/2  

5. a−b/2  

6. a/2  

7. b/2  

8. c/2  

9. AC¯¯¯¯¯  

10. BD¯¯¯¯¯  

11. CE¯¯¯¯¯  

12. DE¯¯¯¯¯