Question

direction: choose the letter of the correct answer and write it on a separate sheet of paper.

1. in the cartesian plane, what is the distance of the point (2, -3) from the origin?

a) √5 b) √13 c) 5 d) 13

1. what is the distance between point a(3, 4) and point b (10, 4)?

a) 4 b) 5 c) 6 d) 7

1. which of the following should be the value of x so that the distance between the points (x, -2) and (12, -2) is 7?

a) 2 b) 3 c) 4 d) 5

1. which of the following equation describes the distance formula?

a) d = √((x₂ - x₁)² + (y₂ - y₁)²) c)d = √((x₂ - x₁)² - (y₂ - y₁)²)
b) d = √((x₂ + x₁)² + (y₂ + y₁)²) d)d = √((x₂ + x₁)² - (y₂ + y₁)²)

1. what is the distance between two points with coordinates (4, 3) and (1,7)?

a) 25 b) 16 c) 5 d) 2

1. what is the distance between point a (4a, 4a) and c (0, a)?

a) 3a b)5a c) 9a d)12a

1. both points d and u are on the fourth quadrant. if the distance between point d and u is 3 units and d is at (2,-1), which of the following are the coordinates of point u?

a) (2,-4) b) (1, -6) c) (4,-1) d) (2,-5)

Explanation:

Step1: Distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step2: Solve problem 1

For the point \((2,-3)\) and the origin \((0,0)\), \(x_1 = 0,y_1 = 0,x_2=2,y_2=-3\).
\(d=\sqrt{(2 - 0)^2+(-3 - 0)^2}=\sqrt{4 + 9}=\sqrt{13}\).

Step3: Solve problem 2

For \(A(3,4)\) and \(B(10,4)\), \(x_1 = 3,y_1 = 4,x_2 = 10,y_2 = 4\).
\(d=\sqrt{(10 - 3)^2+(4 - 4)^2}=\sqrt{49+0}=7\).

Step4: Solve problem 3

For points \((x,-2)\) and \((12,-2)\), \(y_1=-2,y_2=-2\).
\(d=\sqrt{(12 - x)^2+(-2+2)^2}=|12 - x|\). Since \(d = 7\), \(|12 - x|=7\), then \(12 - x=7\) or \(12 - x=-7\).
If \(12 - x=7\), \(x = 5\); if \(12 - x=-7\), \(x=19\).

Step5: Solve problem 4

By the distance formula definition, \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step6: Solve problem 5

For \((4,3)\) and \((1,7)\), \(x_1 = 4,y_1 = 3,x_2 = 1,y_2 = 7\).
\(d=\sqrt{(1 - 4)^2+(7 - 3)^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Step7: Solve problem 6

For \(A(4a,4a)\) and \(C(0,a)\), \(x_1 = 4a,y_1 = 4a,x_2 = 0,y_2 = a\).
\(d=\sqrt{(0 - 4a)^2+(a - 4a)^2}=\sqrt{16a^2+9a^2}=\sqrt{25a^2}=5a\).

Step8: Solve problem 7

Since both \(D\) and \(U\) are in the fourth - quadrant (\(x>0,y<0\)). For \(D(2,-1)\), if \(U(x,y)\) and \(d = 3\).
If \(x = 2\), \(d=\sqrt{(2 - 2)^2+(y + 1)^2}=|y + 1|\). Since \(d = 3\), \(|y+1|=3\), \(y=-4\) or \(y = 2\) (reject \(y = 2\) as \(y<0\)).

Answer:

1. B. \(\sqrt{13}\)
2. D. \(7\)
3. D. \(5\)
4. A. \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)
5. C. \(5\)
6. B. \(5a\)
7. A. \((2,-4)\)