Question

After working with circles and diagonals, JoAnn noticed a pattern. After examining this pattern she concluded that the number of sectors in a circle is always equal to 2 times the number of diagonals drawn. Which statement best justifies her conjecture? JoAnn's conjecture is true because no counterexamples exist. JoAnn's conjecture is false because the sectors are not always equal when drawing more than one diagonal. JoAnn's conjecture is false because when there are 0 diagonals drawn there is still 1 sector. JoAnn's conjecture is false because each diagonal drawn results in a number of sectors equal to the number of diagonals.

Answer 1

JoAnn's conjecture about the number of sectors being double the number of diagonals drawn in a circle is false. This is because drawing no diagonals still results in one sector, and additional diagonals can increase sectors by varying amounts.

Step-by-step explanation:

To address the student's question on JoAnn's conjecture regarding the relationship between the number of diagonals in a circle and the resulting number of sectors, we must understand the properties of circle geometry. JoAnn's conjecture states that the number of sectors in a circle is always equal to 2 times the number of diagonals drawn. However, this conjecture is incorrect. Drawing one diagonal in a circle divides it into 2 sectors, but drawing a second diagonal can either divide one of those sectors into two (if the diagonals are not concurrent), resulting in a total of 3 sectors, or split the circle into 4 sectors if the diagonals cross at the center. Every additional diagonal can cut through existing sectors in various ways, increasing the total number of sectors by varying amounts. Furthermore, drawing no diagonals results in one sector, which is the entire circle. Hence, the correct statement that justifies JoAnn's conjecture being false is that JoAnn's conjecture is false because when there are 0 diagonals drawn, there is still 1 sector.

Answer 2

JoAnn's conjecture is false because it does not hold when there are zero diagonals drawn; there is always at least one sector in a circle regardless of whether any diagonals are present.

Therefore, the correct answer is: option "JoAnn's conjecture is false because when there are 0 diagonals drawn there is still 1 sector".

Step-by-step explanation:

JoAnn has made a conjecture concerning the relationship between the number of sectors formed by the diagonals in a circle and the number of diagonals drawn. She noticed that the number of sectors in a circle is always equal to two times the number of diagonals drawn.

To assess this, consider when there are zero diagonals in a circle; there is still one sector. Therefore, her conjecture fails because with zero diagonals, the formula would suggest that there are zero sectors, which is not true in any circle.

Mathematically, as you draw each diagonal within a circle, you divide the circle into additional parts. However, one diagonal does not simply double the number of sectors that previously existed. Instead, each additional diagonal can intersect with other diagonals creating more sectors, but not in a manner where the count of sectors is strictly twice the number of diagonals.

Therefore, the conjecture is false, and the statement that best justifies this conclusion is: JoAnn's conjecture is false because when there are 0 diagonals drawn there is still 1 sector.