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

Step2: Horizontal distance

For points \(A(3,4)\) and \(B(10,4)\), since \(y_1=y_2 = 4\), \(d=\vert x_2 - x_1\vert=\vert10 - 3\vert = 7\).

Step3: Using distance formula for horizontal line

For points \((x,-2)\) and \((12,-2)\) (since \(y_1=y_2=-2\)), \(d=\vert x - 12\vert\). Given \(d = 7\), then \(\vert x-12\vert=7\), so \(x-12 = 7\) or \(x - 12=-7\). Solving \(x-12 = 7\) gives \(x = 19\), solving \(x - 12=-7\) gives \(x=5\).

Step4: Distance formula recognition

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

Step5: Applying distance formula

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

Step6: Distance formula with variables

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

Step7: Analyzing coordinates in the fourth - quadrant

Since both points are in the fourth quadrant (\(x>0,y<0\)) and for point \(D(2,-1)\), if the distance is \(3\) units and \(x\) - coordinate remains the same (horizontal distance), then \(y=-1-3=-4\) (because moving down along the \(y\) - axis).

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. \((2,-4)\)