Question

1. in the standard (x, y) coordinate plane, what is the length of the line segment that has endpoints (-3, 4) and (5, -6)?

a. 9
b. 2√41
c. 18
d. 20√2
e. 40

1. a triangle has sides of length 4.7 meters and 9 meters. which of the following cannot be the length of the third side, in meters?

a. 5
b. 7
c. 8
d. 11
e. 14

1. what is the center of the circle with equation (x - 3)² + (y + 3)² = 4 in the standard (x, y) coordinate plane?

a. (3, 3)
b. (3, -3)
c. (√3, -√3)
d. (-3, 3)
e. (-√3, √3)

Explanation:

Step1: Apply the distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1=-3,y_1 = 4,x_2=5,y_2=-6\). Then \((x_2 - x_1)=5-(-3)=8\) and \((y_2 - y_1)=-6 - 4=-10\).

Step2: Calculate the distance

Substitute into the formula: \(d=\sqrt{8^2+(-10)^2}=\sqrt{64 + 100}=\sqrt{164}=\sqrt{4\times41}=2\sqrt{41}\)

Step3: Apply the triangle - side length rule

For a triangle with side lengths \(a,b,c\), the triangle inequality states \(|a - b|\lt c\lt a + b\). Here \(a = 4,b = 9\), so \(|9 - 4|=5\) and \(9+4 = 13\). The third - side length \(c\) must satisfy \(5\lt c\lt13\).

Step4: Recall the circle equation formula

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center. For the equation \((x - 3)^2+(y+3)^2 = 4=(x - 3)^2+(y-(-3))^2\), the center is \((h,k)=(3,-3)\)

Answer:

1. B. \(2\sqrt{41}\)
2. E. \(14\)
3. B. \((3,-3)\)