(a) The slope of the line that contains A and B can be found using the formula (y2 - y1) / (x2 - x1). Substituting the given points, we get (0 - 6) / (5 - (-3)) = -6 / 8 = -3/4. So, the slope of the line is -3/4.

(b) The equation of the line can be found using the point-slope form y - y1 = m(x - x1). Substituting the point A(-3,6) and the slope -3/4, we get y - 6 = -3/4(x + 3). Simplifying this, we get 4y - 24 = -3x - 9. Rearranging terms, we get 3x + 4y = 15. The x-intercept is found by setting y = 0, which gives x = 5. The y-intercept is found by setting x = 0, which gives y = 15/4. So, the x-intercept is (5,0) and the y-intercept is (0,15/4).

(c) The midpoint of the segment AB can be found using the formula [(x1 + x2)/2 , (y1 + y2)/2]. Substituting the given points, we get [(-3 + 5)/2 , (6 + 0)/2] = [1, 3]. So, the midpoint is (1,3).

(d) The length of the segment AB can be found using the distance formula sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Substituting the given points, we get sqrt[(5 - (-3))^2 + (0 - 6)^2] = sqrt[64 + 36] = sqrt[100] = 10. So, the length of the segment AB is 10.

(e) The equation of the perpendicular bisector of AB can be found by first finding the slope of the perpendicular line, which is the negative reciprocal of the slope of AB. So, the slope of the perpendicular line is 4/3. Using the point-slope form with the midpoint (1,3) and the slope 4/3, we get y - 3 = 4/3(x - 1). Simplifying this, we get 3y - 9 = 4x - 4. Rearranging terms, we get 4x - 3y = -5.

(f) The equation of the circle for which AB is a diameter can be found using the standard form (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius. The center of the circle is the midpoint of AB, which is (1,3). The radius is half the length of AB, which is 10/2 = 5. Substituting these values, we get (x - 1)^2 + (y - 3)^2 = 25.

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(a) The slope of the line that contains A and B can be found using the formula (y2 - y1) / (x2 - x1). Substituting the given points, we get (0 - 6) / (5 - (-3)) = -6 / 8 = -3/4. So, the slope of the line is -3/4.

(b) The equation of the line can be found using the point-slope form y - y1 = m(x - x1). Substituting the point A(-3,6) and the slope -3/4, we get y - 6 = -3/4(x + 3). Simplifying this, we get 4y - 24 = -3x - 9. Rearranging terms, we get 3x + 4y = 15. The x-intercept is found by setting y = 0, which gives x = 5. The y-intercept is found by setting x = 0, which gives y = 15/4. So, the x-intercept is (5,0) and the y-intercept is (0,15/4).

(c) The midpoint of the segment AB can be found using the formula [(x1 + x2)/2 , (y1 + y2)/2]. Substituting the given points, we get [(-3 + 5)/2 , (6 + 0)/2] = [1, 3]. So, the midpoint is (1,3).

(d) The length of the segment AB can be found using the distance formula sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Substituting the given points, we get sqrt[(5 - (-3))^2 + (0 - 6)^2] = sqrt[64 + 36] = sqrt[100] = 10. So, the length of the segment AB is 10.

(e) The equation of the perpendicular bisector of AB can be found by first finding the slope of the perpendicular line, which is the negative reciprocal of the slope of AB. So, the slope of the perpendicular line is 4/3. Using the point-slope form with the midpoint (1,3) and the slope 4/3, we get y - 3 = 4/3(x - 1). Simplifying this, we get 3y - 9 = 4x - 4. Rearranging terms, we get 4x - 3y = -5.

(f) The equation of the circle for which AB is a diameter can be found using the standard form (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius. The center of the circle is the midpoint of AB, which is (1,3). The radius is half the length of AB, which is 10/2 = 5. Substituting these values, we get (x - 1)^2 + (y - 3)^2 = 25.

Knowee AI · Accepted Answer

(a) The slope of the line that contains A and B can be found using the formula (y2 - y1) / (x2 - x1). Substituting the given points, we get (0 - 6) / (5 - (-3)) = -6 / 8 = -3/4. So, the slope of the line is -3/4.

(b) The equation of the line can be found using the point-slope form y - y1 = m(x - x1). Substituting the point A(-3,6) and the slope -3/4, we get y - 6 = -3/4(x + 3). Simplifying this, we get 4y - 24 = -3x - 9. Rearranging terms, we get 3x + 4y = 15. The x-intercept is found by setting y = 0, which gives x = 5. The y-intercept is found by setting x = 0, which gives y = 15/4. So, the x-intercept is (5,0) and the y-intercept is (0,15/4).

(c) The midpoint of the segment AB can be found using the formula [(x1 + x2)/2 , (y1 + y2)/2]. Substituting the given points, we get [(-3 + 5)/2 , (6 + 0)/2] = [1, 3]. So, the midpoint is (1,3).

(d) The length of the segment AB can be found using the distance formula sqrt[(x2 - x1)^2 + (y2 - y1)^2]. Substituting the given points, we get sqrt[(5 - (-3))^2 + (0 - 6)^2] = sqrt[64 + 36] = sqrt[100] = 10. So, the length of the segment AB is 10.

(e) The equation of the perpendicular bisector of AB can be found by first finding the slope of the perpendicular line, which is the negative reciprocal of the slope of AB. So, the slope of the perpendicular line is 4/3. Using the point-slope form with the midpoint (1,3) and the slope 4/3, we get y - 3 = 4/3(x - 1). Simplifying this, we get 3y - 9 = 4x - 4. Rearranging terms, we get 4x - 3y = -5.

(f) The equation of the circle for which AB is a diameter can be found using the standard form (x - h)^2 + (y - k)^2 = r^2, where (h,k) is the center of the circle and r is the radius. The center of the circle is the midpoint of AB, which is (1,3). The radius is half the length of AB, which is 10/2 = 5. Substituting these values, we get (x - 1)^2 + (y - 3)^2 = 25.

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