Question

the points are the vertices of a figure in the coordinate plane. plot the points and find the perimeter and/or area of the figure.

1. (- 1, - 1), (4, - 1), (4, 2), (- 1, 2); perimeter and area
2. (0, 0), (2, 2), (- 5, 1); perimeter only

topic resource: perimeter and area
find b such that x is the midpoint between a and b.

1. a(9, - 4), x(2, 2)

topic resource: midpoint and distance formulas
solve the word problems.

1. the base of a ladder is 10 feet away from the bottom of a buildings wall. if the ladder makes a 35 angle with the ground, find the length of the ladder to the nearest tenth of a foot.
2. with the same given information as the previous problem, determine how high the ladder can reach up the wall of the building.

Explanation:

41.

Step1: Identify the figure

The points (-1, -1), (4, -1), (4, 2), (-1, 2) form a rectangle. The length \(l\) is the distance between (-1,-1) and (4,-1) and the width \(w\) is the distance between (-1,-1) and (-1,2).

Step2: Calculate the length

Using the distance formula for two - points \((x_1,y_1)\) and \((x_2,y_2)\) \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), for points (-1,-1) and (4,-1), \(x_1=-1,y_1 = - 1,x_2=4,y_2=-1\). Then \(l=\sqrt{(4+1)^2+( - 1 + 1)^2}=\sqrt{25}=5\).

Step3: Calculate the width

For points (-1,-1) and (-1,2), \(x_1=-1,y_1=-1,x_2=-1,y_2 = 2\). Then \(w=\sqrt{(-1 + 1)^2+(2 + 1)^2}=\sqrt{9}=3\).

Step4: Calculate the perimeter

The perimeter of a rectangle \(P = 2(l + w)\), so \(P=2(5 + 3)=16\).

Step5: Calculate the area

The area of a rectangle \(A=l\times w\), so \(A=5\times3 = 15\).

Step1: Use the distance formula

Let \(A=(0,0)\), \(B=(2,2)\), \(C=(-5,1)\). The distance 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: Calculate \(AB\)

For \(A=(0,0)\) and \(B=(2,2)\), \(d_{AB}=\sqrt{(2 - 0)^2+(2 - 0)^2}=\sqrt{4 + 4}=\sqrt{8}=2\sqrt{2}\).

Step3: Calculate \(BC\)

For \(B=(2,2)\) and \(C=(-5,1)\), \(d_{BC}=\sqrt{(-5 - 2)^2+(1 - 2)^2}=\sqrt{49+1}=\sqrt{50}=5\sqrt{2}\).

Step4: Calculate \(CA\)

For \(C=(-5,1)\) and \(A=(0,0)\), \(d_{CA}=\sqrt{(0 + 5)^2+(0 - 1)^2}=\sqrt{25 + 1}=\sqrt{26}\).

Step5: Calculate the perimeter

\(P=d_{AB}+d_{BC}+d_{CA}=2\sqrt{2}+5\sqrt{2}+\sqrt{26}=7\sqrt{2}+\sqrt{26}\approx7\times1.414+5.099=9.898 + 5.099=14.997\approx15.0\).

Step1: Recall the mid - point formula

The mid - point formula between two points \(A=(x_A,y_A)\) and \(B=(x_B,y_B)\) is \(X=(\frac{x_A+x_B}{2},\frac{y_A+y_B}{2})\). Given \(A=(9,-4)\) and \(X=(2,2)\), we have \(\frac{9 + x_B}{2}=2\) and \(\frac{-4 + y_B}{2}=2\).

Step2: Solve for \(x_B\)

From \(\frac{9 + x_B}{2}=2\), we cross - multiply: \(9+x_B=4\), then \(x_B=4 - 9=-5\).

Step3: Solve for \(y_B\)

From \(\frac{-4 + y_B}{2}=2\), we cross - multiply: \(-4 + y_B=4\), then \(y_B=4 + 4=8\).

Answer:

Perimeter: 16, Area: 15

42.