Coordinate Geometry SEC 3 Quiz 2

Explanation
The equation of a line can be determined using the point-slope form, which is y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line. In this case, point B(5,7) lies on the line parallel to AC. The slope of AC can be found using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of two points on the line. In this case, the slope of AC is (0 - 7) / (4 - 5) = -7. Therefore, the slope of the line parallel to AC is also -7. Using point-slope form with point B, the equation of the line is y - 7 = -7(x - 5), which simplifies to y = -7x + 35 + 7, and further simplifies to y = -7x + 42. In the given answer, the equation y = -0.5x + 9.5 is equivalent to y = -1/2x + 9.5, and both represent the equation of the line parallel to AC.

Explanation
The coordinates of the midpoint of a line segment can be found by taking the average of the x-coordinates and the average of the y-coordinates of the endpoints. In this case, the x-coordinate of the midpoint is (-1 + 5)/2 = 2 and the y-coordinate of the midpoint is (5 + 1)/2 = 3. Therefore, the coordinates of the midpoint of AC are (2,3).

Explanation
The equation of BD can be found by using the fact that BD is parallel to the line 5x + y - 2 = 0. Since the line has a slope of -5, the equation of BD can be written in the form y = -5x + c, where c is the y-intercept. To find the value of c, we can substitute the coordinates of point B (-1,5) into the equation. Plugging in the values, we get 5 = -5(-1) + c, which simplifies to 5 = 5 + c. Solving for c, we find c = 0. Therefore, the equation of BD is y = -5x + 0, which simplifies to y = -5x.

Explanation
The equation of BC is y=1.5x-6.5 or y=-6.5+1.5x.

Explanation
The coordinates of point C can be determined by finding the midpoint between points A and B, and then finding the point(s) on the circle with diameter AC that are equidistant from points A and B. In this case, the midpoint between A(2, 6) and B(9, 5) is ((2+9)/2, (6+5)/2) = (5.5, 5.5). Since AB = BC, the distance between A and C is the same as the distance between B and C. Using the distance formula, we can calculate the distance between A and C as √((2-5.5)^2 + (6-5.5)^2) = √(12.25 + 0.25) = √12.5. Therefore, the possible coordinates for C are the points on the circle with diameter AC and a distance of √12.5 from A. These points can be calculated as (5.5 ± √12.5, 5.5). Simplifying, we get (8, -2) and (10, 12). So the coordinates of point C are (8, -2) and (10, 12), and the answer is (8, -2),(10, 12),(8, -2)or(10, 12).

Explanation
The equation of the circle is given in the form (x-a)^2 + (y-b)^2 = r^2, where (a,b) represents the coordinates of the center of the circle. By rearranging the given equation, we can rewrite it as (x-1)^2 + (y-3)^2 = 0. Since the equation is already in the correct form, we can conclude that the center of the circle is located at the point (1,3).

Explanation
The equation of a circle with center (h,k) and radius r is given by (x-h)^2 + (y-k)^2 = r^2. In this case, the center of the circle is (-3,4) and it touches the y-axis, which means the x-coordinate of the center is also the radius. Therefore, the equation of the circle is (x+3)^2 + (y-4)^2 = 3^2, which simplifies to x^2 + y^2 + 6x - 8y + 16 = 0.