Question

1. plot the coordinates on the plane. then determine whether ab and cd are congruent. a(6, - 1), b(1, - 1), c(2, - 3), d(4, - 3)
2. a(3,6), b(3,0), c(-4,5), d(2,5)

Explanation:

Step1: Plot the points

Move along x and y - axes as per coordinates.

Step2: Recall distance formula

\(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step3: Calculate length of AB in first case

Substitute \(x_1 = 6,y_1=-1,x_2 = 1,y_2=-1\) into formula.

Step4: Calculate length of CD in first case

Substitute \(x_1 = 2,y_1=-3,x_2 = 4,y_2=-3\) into formula.

Step5: Compare lengths in first case

Check if \(d_{AB}=d_{CD}\).

Step6: Calculate length of AB in second case

Substitute \(x_1 = 3,y_1=6,x_2 = 3,y_2=0\) into formula.

Step7: Calculate length of CD in second case

Substitute \(x_1=-4,y_1 = 5,x_2 = 2,y_2=5\) into formula.

Step8: Compare lengths in second case

Check if \(d_{AB}=d_{CD}\).

Answer:

1. To plot the points:

• For point A(6, - 1): Start at the origin (0,0). Move 6 units to the right along the x - axis and 1 unit down along the y - axis.
• For point B(1, - 1): Move 1 unit to the right along the x - axis and 1 unit down along the y - axis.
• For point C(2, - 3): Move 2 units to the right along the x - axis and 3 units down along the y - axis.
• For point D(4, - 3): Move 4 units to the right along the x - axis and 3 units down along the y - axis.
• To determine if AB and CD are congruent, we use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
• For AB: Let \(A(x_1,y_1)=(6,-1)\) and \(B(x_2,y_2)=(1,-1)\).
• \(d_{AB}=\sqrt{(1 - 6)^2+(-1+1)^2}=\sqrt{(-5)^2+0^2}=\sqrt{25}=5\).
• For CD: Let \(C(x_1,y_1)=(2,-3)\) and \(D(x_2,y_2)=(4,-3)\).
• \(d_{CD}=\sqrt{(4 - 2)^2+(-3 + 3)^2}=\sqrt{2^2+0^2}=\sqrt{4}=2\).
• Since \(d_{AB}

eq d_{CD}\), AB and CD are not congruent.

1. For points A(3,6), B(3,0), C(-4,5), D(2,5):

• Using the distance formula for AB: Let \(A(x_1,y_1)=(3,6)\) and \(B(x_2,y_2)=(3,0)\).
• \(d_{AB}=\sqrt{(3 - 3)^2+(0 - 6)^2}=\sqrt{0+(-6)^2}=\sqrt{36}=6\).
• Using the distance formula for CD: Let \(C(x_1,y_1)=(-4,5)\) and \(D(x_2,y_2)=(2,5)\).
• \(d_{CD}=\sqrt{(2 + 4)^2+(5 - 5)^2}=\sqrt{6^2+0^2}=\sqrt{36}=6\).
• Since \(d_{AB}=d_{CD}=6\), AB and CD are congruent.