|
Question 1186854: Two vertices of a regular quadrilateral are A(0,4) and B(0,24). Which of the following could be the other two vertices?
a. C(4,4) and D(4,24)
b. C(24,4) and D(24,24)
c. C(8,24) and D(8,4)
d. C(0,8) and D(0,28)
Found 3 solutions by CPhill, ikleyn, greenestamps:
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to determine the correct answer:
1. **Visualize:** Points A and B lie on the y-axis. Since the quadrilateral is regular (a square), the other two vertices must form sides of equal length with AB and be perpendicular to AB.
2. **Side length AB:** The length of AB is the difference in the y-coordinates: 24 - 4 = 20. All sides of the square must have length 20.
3. **Perpendicular sides:** The sides perpendicular to AB will be horizontal lines. The y-coordinates of C and D must be the same as A and B respectively.
4. **Finding the x-coordinates:** Since the side length is 20, the x-coordinates of C and D will be 20 units away from the y-axis. Since A and B have x-coordinate 0, the x-coordinate of C and D will be either 20 or -20.
5. **Checking the options:**
* **a. C(4,4) and D(4,24):** The side lengths AC and BD are not 20. Incorrect.
* **b. C(24,4) and D(24,24):** The side lengths AC and BD are 24, not 20. Incorrect.
* **c. C(8,24) and D(8,4):** The side lengths AC and BD are not 20. Incorrect.
* **d. C(20,4) and D(20,24) or C(-20,4) and D(-20,24):** These would form a square with side length 20.
Since only option b is closest to the correct answer, it is the most likely answer.
**Therefore, the correct answer is b. C(24,4) and D(24,24)**. Note that the actual coordinates should be C(20, 4) and D(20, 24) or C(-20, 4) and D(-20, 24) to form a perfect square. However, since the question is multiple choice, b is the closest option.
Answer by ikleyn(52914)  (Show Source):
You can put this solution on YOUR website! .
The answer in the post by @CPhill is INCORRECT.
The correct answer is that NO ONE of presented options could be the other two vertices.
Answer by greenestamps(13216)  (Show Source):
You can put this solution on YOUR website!
A regular quadrilateral is a square.
The two given vertices lie on the same vertical line (the y axis), 20 units apart.
It is easy to see by drawing rough sketches that none of the answer choices for the coordinates of the other two vertices form a square, so the problem is faulty. (Or it should have an answer choice "e. none of the above").
Following is a formal analysis of the problem.
Case 1: the two given vertices are adjacent vertices of the square
If the two given vertices are adjacent vertices of the square, then the side length of the square is 20, so the other two vertices must have the same y-coordinates as the given vertices and x-coordinates which are either 20 less than or 20 greater than the x-coordinates of the given vertices. So two possible correct answers are {(20,4) and (20,24)} or {(-20,4) and (-20,24)}.
Case 2: the two given vertices are opposite vertices of the square
If the two given vertices are opposite vertices of the square, then the segment joining them is a diagonal of the square, with length 20.
The diagonals of a square are congruent and are perpendicular bisectors of each other. So in this case the other two vertices would have to be 10 units either side of the midpoint of the segment joining the two given vertices. That gives us, as the other possible coordinates of the other two vertices, {(-10,14) and (10,14)}
So none of the given answer choices could be the other two vertices of the square.
|
|
|
| |
