A, B centers of non-congruent circles intersecting at C, D. Find most precise name for ACBD....

[Faye]

Question 3

A and B are centers of two non-congruent circles which intersect at points C and D.

(i) Determine the most precise name for quadrilateral ACBD. Justify your answer.
(ii) Identify the relationship between AB and CD. Use mathematics to explain your answer.

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If anyone could offer advice with this question I would appreciate it. Thank you!

 

[pka]

Question 3

A and B are centers of two non-congruent circles which intersect at points C and D.

(i) Determine the most precise name for quadrilateral ACBD. Justify your answer.
(ii) Identify the relationship between AB and CD. Use mathematics to explain your answer.

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If anyone could offer advice with this question I would appreciate it. Thank you!

Here is the answer. But you need to post your justification so we can check it.

 

[MarkFL]

Let's look at a diagram:


Now, consider what happens if $\overline{AB}$ changes length, or the radius of either circle changes...what are your thoughts?

 

[Faye]

The parallelogram looks like a kite. If the length of AB got longer or the radius changed, the size of the parallelogram would change. Also, the adjacent sides are equal. There are two pairs of equal sides. I know there is a relationship between the radius and the relationship between AB and CD but I don’t know how to justify it. AB and CD would create a right angle where they meet??

 

[Faye]

The parallelogram looks like a kite. If the length of AB got longer or the radius changed, the size of the parallelogram would change. Also, the adjacent sides are equal. There are two pairs of equal sides. I know there is a relationship between the radius and the relationship between AB and CD but I don’t know how to justify it. AB and CD would create a right angle where they meet??

Also, if AB and CD length or the radius, the lengths of the sides change? Is that the answer I’m looking for? Also CD is perpendicular to AB

 

[pka]

The parallelogram looks like a kite. If the length of AB got longer or the radius changed, the size of the parallelogram would change. Also, the adjacent sides are equal. There are two pairs of equal sides. I know there is a relationship between the radius and the relationship between AB and CD but I don’t know how to justify it. AB and CD would create a right angle where they meet??

A kite & a parallelogram are not the same. In the above you seem to imply that they are the same. In fact both are quadrilaterals but there the similarities stop. In a parallelogram opposite sides are parallel and congruent. Whereas in a kite pairs of adjacent sides are congruent and the diagonals are perpendicular. Look at $\displaystyle \overline{AB}$ is the line segment of the centers. The line segment $\displaystyle \overline{CD}$ is the common chord in both circles. As such we see that $\displaystyle \overline{AB}\bot\overline{CD}$ AND those two are the diagonals of the kite $\displaystyle {ACBD}$ in which $\displaystyle \overline{AC}\cong\overline{AD}$ & $\displaystyle \overline{BC}\cong\overline{BD}$.

 

[Faye]

A kite & a parallelogram are not the same. In the above you seem to imply that they are the same. In fact both are quadrilaterals but there the similarities stop. In a parallelogram opposite sides are parallel and congruent. Whereas in a kite pairs of adjacent sides are congruent and the diagonals are perpendicular. Look at $\displaystyle \overline{AB}$ is the line segment of the centers. The line segment $\displaystyle \overline{CD}$ is the common chord in both circles. As such we see that $\displaystyle \overline{AB}\bot\overline{CD}$ AND those two are the diagonals of the kite $\displaystyle {ACBD}$ in which $\displaystyle \overline{AC}\cong\overline{AD}$ & $\displaystyle \overline{BC}\cong\overline{BD}$.

Thank you for noticing that. I meant to say it was a quadrilateral. I was so exhausted

 

[MarkFL]

What happens if $\overline{AB}$ and $\overline{CD}$ don't actually intersect? It seems we could get a concave quadrilateral...

 

[Dr.Peterson]

In which case,

it can still be called a kite, though it is also called a dart.

 

[Faye]

A kite & a parallelogram are not the same. In the above you seem to imply that they are the same. In fact both are quadrilaterals but there the similarities stop. In a parallelogram opposite sides are parallel and congruent. Whereas in a kite pairs of adjacent sides are congruent and the diagonals are perpendicular. Look at $\displaystyle \overline{AB}$ is the line segment of the centers. The line segment $\displaystyle \overline{CD}$ is the common chord in both circles. As such we see that $\displaystyle \overline{AB}\bot\overline{CD}$ AND those two are the diagonals of the kite $\displaystyle {ACBD}$ in which $\displaystyle \overline{AC}\cong\overline{AD}$ & $\displaystyle \overline{BC}\cong\overline{BD}$.

Amazing, I just learned so much! Thank you.