Question

1. in the cartesian plane, what is the distance of the point (5,6) from the origin?

a) 61
b) 11
c)√61
d) 11

1. what is the distance between point a(-3,1) and point b(11,1)?

a) 11
b) 12
c) 13
d) 14

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

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

1. which of the following describes the distance formula?

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

Explanation:

Step1: Distance from point \((x,y)\) to origin \((0,0)\)

The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here \(x_1 = 0,y_1 = 0,x_2=5,y_2 = 6\). So \(d=\sqrt{(5 - 0)^2+(6 - 0)^2}=\sqrt{25 + 36}\).

Step2: Simplify the expression

\(\sqrt{25+36}=\sqrt{61}\).

Step3: Distance between \(A(-3,1)\) and \(B(11,1)\)

Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \(x_1=-3,y_1 = 1,x_2=11,y_2 = 1\). Then \(d=\sqrt{(11-(-3))^2+(1 - 1)^2}=\sqrt{(14)^2+0}=\sqrt{196}=14\).

Step4: Distance between \((2,-2)\) and \((2,y)\)

Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \(x_1 = 2,y_1=-2,x_2 = 2,y_2=y\). Given \(d = 7\), so \(7=\sqrt{(2 - 2)^2+(y-(-2))^2}\). Since \((2 - 2)^2=0\), then \(7=\sqrt{(y + 2)^2}\), which means \(|y+2|=7\). Solving \(y+2=7\) gives \(y = 5\) and \(y+2=-7\) gives \(y=-9\). But \(y = 5\) is in the options.

Step5: Recall the distance formula

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

Answer:

1. C. \(\sqrt{61}\)
2. D. \(14\)
3. D. \(5\)
4. A. \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)