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Math  /  Geometry

QuestionThe points H(8,2),I(−1,8),J(−3,5)\mathrm{H}(8,2), \mathrm{I}(-1,8), \mathrm{J}(-3,5) , and K(6,−1)\mathrm{K}(6,-1) form rectangle HIJK. Plot the points then click the "Graph Quadrilateral" button. Then find the area of the rectangle.

Studdy Solution

STEP 1

1. The points H(8,2)\mathrm{H}(8,2), I(−1,8)\mathrm{I}(-1,8), J(−3,5)\mathrm{J}(-3,5), and K(6,−1)\mathrm{K}(6,-1) form a rectangle.
2. The area of a rectangle can be found by multiplying its length and width.
3. To find the length and width, we need to calculate the distances between points determining the sides of the rectangle.
4. The distance between two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by the formula: d=(x2−x1)2+(y2−y1)2d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.

STEP 2

1. Verify that the given points form a rectangle by checking distances and right angles.
2. Calculate the lengths of the sides of the rectangle.
3. Compute the area of the rectangle using the lengths of the sides.

STEP 3

Calculate the distance between points H(8,2)\mathrm{H}(8,2) and I(−1,8)\mathrm{I}(-1,8).
dHI=((−1)−8)2+(8−2)2=(−9)2+62=81+36=117=313 d_{\mathrm{HI}} = \sqrt{((-1)-8)^2 + (8-2)^2} = \sqrt{(-9)^2 + 6^2} = \sqrt{81 + 36} = \sqrt{117} = 3\sqrt{13}

STEP 4

Calculate the distance between points I(−1,8)\mathrm{I}(-1,8) and J(−3,5)\mathrm{J}(-3,5).
dIJ=((−3)−(−1))2+(5−8)2=(−2)2+(−3)2=4+9=13 d_{\mathrm{IJ}} = \sqrt{((-3)-(-1))^2 + (5-8)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}

STEP 5

Calculate the distance between points J(−3,5)\mathrm{J}(-3,5) and K(6,−1)\mathrm{K}(6,-1).
dJK=(6−(−3))2+(−1−5)2=(9)2+(−6)2=81+36=117=313 d_{\mathrm{JK}} = \sqrt{(6-(-3))^2 + (-1-5)^2} = \sqrt{(9)^2 + (-6)^2} = \sqrt{81 + 36} = \sqrt{117} = 3\sqrt{13}

STEP 6

Calculate the distance between points K(6,−1)\mathrm{K}(6,-1) and H(8,2)\mathrm{H}(8,2).
dKH=(8−6)2+(2−(−1))2=(2)2+(3)2=4+9=13 d_{\mathrm{KH}} = \sqrt{(8-6)^2 + (2-(-1))^2} = \sqrt{(2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13}

STEP 7

Verify that the points form a rectangle by checking the distances. The opposite sides should be equal:
dHI=dJK=313 d_{\mathrm{HI}} = d_{\mathrm{JK}} = 3\sqrt{13} dIJ=dKH=13 d_{\mathrm{IJ}} = d_{\mathrm{KH}} = \sqrt{13}
Since the opposite sides are equal, we can confirm that HIJK forms a rectangle.

STEP 8

Identify the lengths of the rectangle's sides: - Length: 3133\sqrt{13} - Width: 13\sqrt{13}

STEP 9

Calculate the area of the rectangle using the formula for the area of a rectangle: Area=Length×Width\text{Area} = \text{Length} \times \text{Width}.
Area=(313)×(13)=3×13=39 \text{Area} = (3\sqrt{13}) \times (\sqrt{13}) = 3 \times 13 = 39
The area of the rectangle HIJK is 3939 square units.

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