Answer:
CD = 6.5 cm
Step-by-step explanation:
Shape ABCDEFGH is made up of trapezoid BCDE and rectangle AFGH.
Line segment CD is one of the bases of trapezoid BCDE.
To find the length of CD, we first need to determine the area of trapezoid BCDE, along with the length of base BE and the height of the trapezoid. Then, we can substitute these values into the formula for the area of a trapezoid and solve for the length of base CD.
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Area of trapezoid BCDE
To find the area of trapezoid BCDE, we can subtract the area of rectangle AFGH from the given area of shape ABCDEFGH.
The area of a rectangle is the product of its width and height. In this case, the width of rectangle AFGH is HG = 28 cm and its height is FG = 12 cm. Therefore, its area is:
[tex]\textsf{Area of rectangle $AFGH$} = 28 \times 12\\\\\textsf{Area of rectangle $AFGH$} = 336\; \sf cm^2[/tex]
Given that the area of shape ABCDEFGH is 434 cm², then:
[tex]\textsf{Area of trapezoid $BCDE$} = 434 - 336 \\\\\textsf{Area of trapezoid $BCDE$} = 98\; \sf cm^2[/tex]
Therefore, the area of trapezoid BCDE is 98 cm².
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Base BE of trapezoid BCDE
Base BE of trapezoid BCDE is equal to the width of rectangle AFGH less the lengths of line segments AB and EF:
[tex]\textsf{Base $BE$} = HG - AB - EF\\\\\textsf{Base $BE$} = 28 -5-5\\\\\textsf{Base $BE$} = 18 \; \sf cm[/tex]
Therefore, the length of base BE of trapezoid BCDE is 18 cm.
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Height of trapezoid BCDE
The height (h) of trapezoid BCDE is the height of the figure ABCDEFGH less the height of rectangle AFGH:
[tex]h = 20 - 12\\\\h = 8\; \sf cm[/tex]
Therefore, the height of trapezoid BCDE is 8 cm.
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Length of CD
The formula for the area of a trapezoid is:
[tex]\boxed{\begin{array}{l}\underline{\textsf{Area of a trapezoid}}\\\\A=\dfrac{1}{2}(a+b)h\\\\\textsf{where:}\\ \phantom{ww}\bullet\;\textsf{$A$ is the area.}\\ \phantom{ww}\bullet\;\textsf{$a$ and $b$ are the parallel sides (bases).}\\\phantom{ww}\bullet\;\textsf{$h$ is the height (perpendicular to the bases).}\end{array}}[/tex]
In this case:
- A = 98 cm²
- a = CD
- b = BE = 18 cm
- h = 8 cm
Substitute the values into the formula and solve for CD:
[tex]98=\dfrac{1}{2}(CD+18)\cdot 8\\\\\\98=4(CD+18)\\\\\\\dfrac{98}{4}=CD+18\\\\\\24.5=CD+18\\\\\\CD=24.5-18\\\\\\CD=6.5\; \sf cm[/tex]
Therefore, the length of CD is:
[tex]\Large\boxed{\boxed{CD=6.5\; \sf cm}}[/tex]
