Question

1-3 lesson quiz-exit ticket. items 1 - 3 refer to the graph shown. show work! 1. what is the midpoint of ab? 2. what is the length of ab? round your answer to the nearest hundredth. 3. what are the coordinates of the point 3/4 of the way from a to b? 4. which of the following represent the distance formula? select all that apply. a d = √((x₁ - x₂)²+(y₁ - y₂)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: Identify coordinates

Let \(A(-5,-4)\) and \(B(-3,3)\)

Step2: Find mid - point of \(AB\)

The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). So, \(\frac{-5+( - 3)}{2}=\frac{-8}{2}=-4\) and \(\frac{-4 + 3}{2}=-\frac{1}{2}\). The mid - point is \((-4,-\frac{1}{2})\)

Step3: Find length of \(AB\)

The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1=-5,y_1 = - 4,x_2=-3,y_2 = 3\). Then \(d=\sqrt{(-3-( - 5))^2+(3-( - 4))^2}=\sqrt{(2)^2+(7)^2}=\sqrt{4 + 49}=\sqrt{53}\approx7.28\)

Step4: Find the point \(\frac{3}{4}\) of the way from \(A\) to \(B\)

The formula for the point \(P\) that divides the line segment from \(A(x_1,y_1)\) to \(B(x_2,y_2)\) in the ratio \(m:n\) is \(P=(\frac{mx_2+nx_1}{m + n},\frac{my_2+ny_1}{m + n})\). Here \(m = 3,n = 1\). So \(x=\frac{3\times(-3)+1\times(-5)}{3 + 1}=\frac{-9-5}{4}=\frac{-14}{4}=-\frac{7}{2}\) and \(y=\frac{3\times3+1\times(-4)}{3 + 1}=\frac{9-4}{4}=\frac{5}{4}\). The point is \((-\frac{7}{2},\frac{5}{4})\)

Step5: Select distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) and also \(d=\sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}\) since \((a)^2=|a|^2\). So the correct options are A, B, D

Answer:

1. \((-4,-\frac{1}{2})\)
2. \(7.28\)
3. \((-\frac{7}{2},\frac{5}{4})\)
4. A. \(d=\sqrt{(x_1 - x_2)^2+(y_1 - y_2)^2}\), B. \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), D. \(d=\sqrt{|x_2 - x_1|^2+|y_2 - y_1|^2}\)