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4 years ago

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ABCD is a trapezoid with median EF. a. AB = 15, DC = 11, then EF = ? b. EF = 14, DC = 10, then AB = ? c. EF = 18, AB = (5n - 9),

and DC = (2n + 3). Find n, AB, and DC. d. EF = (2y + 4), AB = (5y + 2), and DC = (-3y + 8). Find y, EF, AB, and DC.

1 answer:

Reptile [31]4 years ago

5 0

The median of the trapezoid is the average of the bases. If we draw the trapezoid that is being described in this item, we will deduce that AB and DC are the bases and EF is given to be the median.

For this item,
EF = (AB + CD) / 2

Part A:
EF = (15 + 11) / 2 = 13

Part B:
AB = 2EF - CD
AB = (2)(14) - 10 = 18

Part C:
18 = ((5n - 9) + (2n + 3))/2
18 = (7n - 6) / 2
n = 6

Part D:
2y + 4 = ((5y + 2) + (-3y + 8))/2

y = 1

EF = 2(1) + 4 = 6
AB = (5(1) + 2 = 7
AB = -3(1) + 8 = 5

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The Perimeter of the full shape is the sum of the lengths of the edges of the parts. For convenience in referencing them, we'll call the large curve " $curve_{big}$ " and the three smaller curves " $curve_1$ " " $curve_2$ " " $curve_3$ " in order from left to right.

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Since values in the diagram are diameters, use the formula for the Perimeter of a circle $P=\pi d$ (where d is the diameter).

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Note that for each curve, the curve only represents half of a circle. So, to find the length of each curve, we'll need half of the full perimeter of each circle.

So for instance: $Length(curve_{big})=\frac{1}{2} \pi d_{big}$

Substituting back into the main equation above:

$P_{overall} = Length(curve_{big})+Length(curve_1)+Length(curve_2)+Length(curve_3)$ $P_{overall}=\frac{1}{2} \pi d_{big} + \frac{1}{2} \pi d_{1} + \frac{1}{2} \pi d_{2} + \frac{1}{2} \pi d_{3}$

Note that all terms have common factors of "one-half" and "pi" in them. These can be factored out:

$P_{overall}=\frac{1}{2} \pi (d_{big} + d_{1} + d_{2} +d_{3})$

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Substituting and simplifying (in terms of pi):

$P_{overall}=\frac{1}{2} \pi ( (12cm) + (4cm) + (4cm) + (4cm) )\\P_{overall}=\frac{1}{2} \pi ( 24cm)\\P_{overall}=12 \pi cm$

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Interesting for this problem, since the diameters of the 3 small curves formed the diameter of the large curve $d_{1} + d_{2} + d_{3} =d_{big}$ , one could make a different substitution into one of our formulas above:

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The number of smaller circles doesn't need to be 3 either... as long as it goes the full distance across, right along the diameter.

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