https://file.aiquickdraw.com/mathbot/file/1728704591774g9gki0fq.jpg

Let's tackle each question in this image one at a time.

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**Question 15**  
Points \( A \), \( B \), and \( C \) do not lie on the same line. The distance from \( A \) to \( B \) is 4 units, and the distance from \( B \) to \( C \) is 5 units. The task is to find the largest possible integer distance between points \( A \) and \( C \).

To find the maximum distance, we should assume points \( A \), \( B \), and \( C \) form a triangle. Using the triangle inequality, the longest side \( AC \) would occur when \( \angle ABC \) is \( 180^\circ \), forming a straight line. The distance \( AC \) can be calculated by adding \( AB \) and \( BC \):  
\[
AC = AB + BC = 4 + 5 = 9
\]
Thus, the largest possible integer distance between points \( A \) and \( C \) is **9**.

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**Question 16**  
In the figure, two tangent segments, \( AP \) and \( BP \), are drawn to a circle with center \( C \). The measure of arc \( ADB \) is \( 240^\circ \). We need to determine which statement is true.

Since the total circle measures \( 360^\circ \), the remaining arc \( APB \) (opposite to \( ADB \)) is:
\[
360^\circ - 240^\circ = 120^\circ
\]
This angle at the center implies that the angle \( \angle APB \), which is half of \( APB \), is:
\[
\angle APB = \frac{120^\circ}{2} = 60^\circ
\]
From this, \( \triangle APB \) forms an isosceles triangle (since \( AP \) and \( BP \) are tangents of equal length). Therefore, the best answer here is **C) \( \triangle ABP \) is an equilateral triangle**.

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**Question 17**  
The equations \( y = 3x - 4 \) and \( 2x + y = 1 \) each describe a line in the \( xy \)-plane that includes a diameter of a circle. We are to find the equation of this circle given that it includes the point \((-2, -5)\).

1. Rewrite the second equation in terms of \( y \):
   \[
   y = -2x + 1
   \]
2. The midpoint of the diameter is the point where these two lines intersect. Setting \( 3x - 4 = -2x + 1 \):
   \[
   5x = 5 \Rightarrow x = 1
   \]
   Substitute \( x = 1 \) back into \( y = 3x - 4 \):
   \[
   y = 3(1) - 4 = -1
   \]
   So, the center of the circle is \( (1, -1) \).

3. To find the radius squared, calculate the distance from the center \( (1, -1) \) to the point \( (-2, -5) \):
   \[
   \text{Distance}^2 = (1 + 2)^2 + (-1 + 5)^2 = 3^2 + 4^2 = 9 + 16 = 25
   \]
Thus, the equation of the circle is:
\[
(x - 1)^2 + (y + 1)^2 = 25
\]
The correct answer is **D**.

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**Question 18**  
Mr. Johnson collected salary data from 20 randomly selected employees. The median salary was \( 37,500 \) while the average was \( 49,500 \). Which choice best explains the discrepancy?

The average salary is much higher than the median, which usually indicates outliers on the high end. Therefore, the correct answer is **B) One or more very high salaries pulled the average up**.

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Would you like further explanations or have any questions?

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### Related Questions

1. How does the triangle inequality affect the maximum possible distance in a triangle?
2. What is the significance of tangents forming an isosceles triangle in circles?
3. How do you find the intersection point of two lines in the coordinate plane?
4. Why do high outliers increase the average more than the median?
5. How can we determine if a triangle is equilateral based on angle and side properties?

### Tip
In geometry problems involving circles, always consider whether tangents or diameters could create special triangles or angles, as this can simplify complex problems.

Points A, B, and C do not lie on the same line. If the distance from A to B is 4 units, and the distance from B to C is 5 units, what is the largest possible integer distance between points A and C? Additionally, two tangent segments are drawn to a circle with a given arc measure, and equations of lines describe a circle diameter in the xy-plane.

Explore a series of geometry problems involving triangle inequality, tangent circles, and the equation of a circle. The solutions cover key concepts such as the maximum possible distance between points, the properties of tangents, and finding the equation of a circle in the xy-plane.

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