Question

Could someone explain to me what I did wrong? I wrote my answer in the essay part, But, I want to know how I would have answered this correctly so I know for next time what I did wrong.

Answer 1

I think you chose the wrong tact to answer this question. Well the wrong approach. What I would have done was prove that the angles at all the midpoints were 90 degrees and used that to demonstrate that it was a rectangle.

Answer 2

Could you further explain @GiantRobot

Answer 3

A rectangle as I recently realized is a quadrilateral with all four angle at 90 degrees. Due to the nature of quadrilaterals, all you need to do is to prove two of the internal angles are 90 degrees.

I'm not sure how to write it out since I have never actually done something like that, no experience.

Answer 4

EDIT: Sorry forgot to state that those two angles cannot be opposite each other.

Answer 5

You need to prove that the quadrilateral with vertices that are midpoints of the rhombus is a rectangle.
A rectangle is a parallelogram with ONE right angle.
If a parallelogram has one right angle, then all angles are right angles, and the parallelogram is a rectangle.

This is why. A parallelogram has congruent opposite angles.
If one angle of a parallelogram is a right angle, then the opposite angle is also a right angle. Those two angles add up to 180 degrees. The sum of the measures of the angles of a quadrilateral is 360. If two angles add up to 180, there are 180 degrees left for the other two angles. Since the other two angles are opposite angles, they are congruent, so each one measures 90, so all angles measure 90.

Answer 6

All the above is just telling you that to prove a quadrilateral is a rectangle, you need to show these two things:
1. The quadrilateral is a parallelogram.
2. One angle is a right angle.

Answer 7

Could you help me do that? I haven't done this in a while, and I'm trying to review for finals

Answer 8

Now you need a way of proving that a quadrilateral is a parallelogram.
One way to do it is using the following theorem:
If in a quadrilateral, both pairs of opposite sides are congruent, then the quadrilateral is a parallelogram.

Answer 9

Use the coordinates of the midpoints of the sides of the rhombus, and find the lengths of all sides of the quadrilateral we want to prove is a parallelogram.

Answer 10

Look at the top horizontal side:
Coordinates: (-a, b) and (a, b)
What is the length of that side?
Since both endpoints are on a horizontal line, you just need to subtract the x-coordinates and take the absolute value.
What is |-a - a| ?

Answer 11

I have no idea:// I'm horrible at this

Answer 12

@quickstudent

Answer 13

Sorry, I've just started learning this and not that familiar yet :(

Answer 14

Find the lengths of the 4 sides of the quadrilateral, and show that each pair of opposite sides is congruent. That proves the quadrilateral is a parallelogram.
Then find the slopes of two consecutive sides, and show they are negative reciprocals. That shows that one angle is a right angle. That is all you need to do to prove the quadrilateral is a rectangle.

Answer 15

@bmk614

Answer 16

The distance between points (-a, b) and (a, b) is: |-a - a| = |-2a| = 2a
The distance between points (-a, -b) and (a, -b) is |-a - a| = |-2a| = 2a
The top and bottom sides of the quadrilateral are congruent.

The distance between points (-a, b) and (-a, -b) is: |b - (-b)| = |b + b| = |2b| = 2b
The distance between points (a, b) and (a, -b) is |b - (-b)| = |b + b| = |2b| = 2b
The left and right sides of the quadrilateral are congruent.

Since the quadrilateral has two pairs of opposite sides congruent, the quadrilateral is a parallelogram.

Answer 17

Now we need to prove that the parallelogram is a rectangle.
All we need to do is to show that two consecutive sides are perpendicular.

Let' use the top side and the right side.

The slope of the top side is \(\dfrac{b - b}{-a - a} = \dfrac{0}{-2a} = 0\)

The slope of the top side is 0, so the top side is horizontal.

Now we find the slope of the right side.

\(\dfrac{-b - b}{a - a} = \dfrac{-2b}{0}\)

Since the slope of the right side involves division by zero, that means the slope is undefined, and an undefined slope means a vertical line.

The top side of the parallelogram is horizontal, and the right side is vertical. The two sides are perpendicular forming right angles when they intersect. That means that the vertex of the parallelogram has a right angle. Since we have a parallelogram with one right angle, it is a rectangle.