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Углы при одном из оснований трапеции равны $80°$ и $10°$ , а отрезки, соединяющие середины противоположных сторон трапеции, равны $20$ и $17$ . Найдите основания трапеции.

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Find derivatives of radical functions

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Q. Углы при одном из оснований трапеции равны

80°

и

10°

, а отрезки, соединяющие середины противоположных сторон трапеции, равны

20

и

17

. Найдите основания трапеции.

Identify Properties and Information: Identify the properties of the trapezoid and the given information. $\newline$ We have a trapezoid with one pair of angles given as $80^\circ$ and $10^\circ$ . The segments connecting the midpoints of the opposite sides (also known as the mid-segment or midline) are given as $20$ and $17$ . The mid-segment of a trapezoid is parallel to the bases and its length is the average of the lengths of the bases.
Set Up Equation: Use the properties of the mid-segment to set up an equation. Let the lengths of the two bases be $a$ and $b$ , with $a$ being the longer base. The mid-segment's length is the average of the two bases, so we have the equation $(a + b)/2 = 20$ .
Solve for Base Length: Solve the equation for one of the bases. $\newline$ We can express $b$ in terms of $a$ by rearranging the equation: $b = 40 - a$ .
Apply Second Mid-Segment Properties: Apply the properties of the trapezoid to the second mid-segment. $\newline$ The second mid-segment, which is $17$ , is not parallel to the bases. It is the segment that connects the midpoints of the non-parallel sides (legs) of the trapezoid. This segment forms two similar right triangles within the trapezoid, one on each side.
Find Ratio of Sides: Use the angles to find the ratio of the sides in the similar right triangles. $\newline$ The angles of $80^\circ$ and $10^\circ$ at the base create two right triangles with the angles $80^\circ$ , $10^\circ$ , and $90^\circ$ , and the other with angles $10^\circ$ , $80^\circ$ , and $90^\circ$ . The ratio of the sides opposite to the $10^\circ$ and $80^\circ$ angles will be the same as the ratio of the sine of these angles. However, this step involves a trigonometric approach that is not necessary for solving the problem, as we can use the properties of the mid-segment directly. Therefore, this step is a misstep.

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