Question

1. the equation of a circle is ((x + 3)^2 + (y - 1)^2 = 36). what is the distance from the center to the point ((1, 1))?

options: a. 8, b. 6, c. 2, d. 4

1. what is the equation of a circle with center at ((2, -5)) and radius 3?

options: a. ((x - 2)^2 + (y + 3)^2 = 9), b. ((x + 2)^2 + (y - 3)^2 = 9), c. ((x - 2)^2 + (y + 5)^2 = 9), d. ((x + 2)^2 + (y - 5)^2 = 3)

1. what variable represents the horizontal difference between two points in the distance formula?

options: a. h, b. s, c. (delta x), d. (delta y)

1. which of the following least describes a rotation?

options: a. a flip across a line, b. a turn around a point, c. an movement in space, d. a shift straight to the side

1. two triangles are congruent by sss. what does this imply about their angles?

options: a. their angles must be supplementary, b. their angles must add up to 180 degrees, c. their angles must be complementary, d. their corresponding angles are equal

1. how do you prove that two lines are perpendicular using their slopes?

options: a. the sum of their slopes is 0, b. the difference of their slopes is 1, c. the product of their slopes is -1, d. the slopes are equal

1. an interior designer is creating a quadrilateral patio with vertices at (a(0, 0)), (b(0, 2)), (c(4, -1)), and (d(2, -4)). which property can prove that the patio is a parallelogram?

options: a. all angles are right angles, b. opposite sides are congruent, c. opposite sides are perpendicular, d. diagonals are congruent

Explanation:

Question 36 (Circle Equation and Distance)

Step1: Identify Circle Center

The standard circle equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center. For \((x + 3)^2 + (y - 1)^2 = 36\), rewrite as \((x - (-3))^2 + (y - 1)^2 = 6^2\). So center is \((-3, 1)\).

Step2: Apply Distance Formula

Distance between \((-3, 1)\) and \((1, 5)\) is \(d = \sqrt{(1 - (-3))^2 + (5 - 1)^2}\). Calculate: \(1 - (-3) = 4\), \(5 - 1 = 4\). Then \(d = \sqrt{4^2 + 4^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\)? Wait, no—wait, the options: a. 8, b. 6, c. 2, d. 4. Wait, maybe I misread the point. Wait, the problem says "distance from the center to the point (1, 5)"? Wait, center is \((-3,1)\), point is \((1,5)\). So \(x\)-difference: \(1 - (-3) = 4\), \(y\)-difference: \(5 - 1 = 4\). Wait, but the circle equation has \(r^2 = 36\), so \(r = 6\). Wait, maybe the point is (1,1)? Wait, the user's image: "the distance from the center to the point (1, )"—maybe a typo, but assuming center \((-3,1)\), point (1,1). Then \(x\)-difference: \(1 - (-3) = 4\), \(y\)-difference: \(1 - 1 = 0\). Distance is 4? No, wait, if center is \((-3,1)\), point (1,1): distance is \(|1 - (-3)| = 4\)? No, distance formula: \(\sqrt{(1 - (-3))^2 + (1 - 1)^2} = \sqrt{16 + 0} = 4\). But the options have d. 4. Wait, maybe the point is (1,1). So step1: center \((h,k) = (-3,1)\). Step2: distance to (1,1): \(d = \sqrt{(1 - (-3))^2 + (1 - 1)^2} = \sqrt{16 + 0} = 4\).

Step1: Recall Circle Equation

Standard form: \((x - h)^2 + (y - k)^2 = r^2\), where \((h,k)\) is center, \(r\) is radius.

Step2: Substitute Values

Center \((2, -5)\), so \(h = 2\), \(k = -5\), radius \(r = 3\). Substitute: \((x - 2)^2 + (y - (-5))^2 = 3^2\) → \((x - 2)^2 + (y + 5)^2 = 9\). Wait, options: a. \((x - 2)^2 + (y + 3)^2 = 81\) (r=9, no), b. \((x + 2)^2 + (y - 3)^2 = 9\) (center wrong), c. \((x - 2)^2 + (y + 5)^2 = 9\) (yes), d. \((x + 2)^2 + (y - 3)^2 = 3\) (wrong). So correct is c.

Step1: Recall Distance Formula

Distance between \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\). The horizontal difference is \(x_2 - x_1\), denoted as \(\Delta x\) (delta x). So the variable is \(\Delta x\) (option c: \(\Delta x\)).

Answer:

d. 4

Question 37 (Circle Equation with Center and Radius)