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Zorluk: OrtaQuadrilaterals and Polygons

An isosceles trapezoid has parallel base lengths of 1010 and 2626, and an altitude of 1515. A line segment connects the midpoints of the two non-parallel legs, dividing the figure into two smaller trapezoids. What is the area of the larger of these two smaller trapezoids?

Cevap: 165

Cevap

165
The midsegment of a trapezoid connects the midpoints of the non-parallel legs, and its length is the average of the two parallel bases: 10+262=18\frac{10 + 26}{2} = 18. Because the line connects midpoints, it also bisects the altitude, making the height of each smaller trapezoid equal to 152=7.5\frac{15}{2} = 7.5. The larger of the two resulting trapezoids has bases of lengths 1818 and 2626. Using the trapezoid area formula Area=b1+b22×h\text{Area} = \frac{b_1 + b_2}{2} \times h, we obtain 18+262×7.5=22×7.5=165\frac{18 + 26}{2} \times 7.5 = 22 \times 7.5 = 165.

Adım Adım Çözüm

1
Calculate the length of the midsegment connecting the midpoints of the non-parallel legs.
The midsegment length is 1818.
The midsegment of a trapezoid is parallel to the bases and its length equals the average of the two base lengths: 10+262=18\frac{10 + 26}{2} = 18.
2
Determine the height of the smaller subtrapezoid.
The height of the subtrapezoid is 7.57.5.
The segment connecting the midpoints of the legs bisects the overall altitude of 1515, giving a height of 152=7.5\frac{15}{2} = 7.5 for each subtrapezoid.
3
Compute the area of the larger subtrapezoid.
The area is 165165.
The larger subtrapezoid is bounded by the midsegment (length 1818) and the bottom base (length 2626). Applying the trapezoid area formula yields Area=18+262×7.5=22×7.5=165\text{Area} = \frac{18 + 26}{2} \times 7.5 = 22 \times 7.5 = 165.

Anahtar Kavram

Trapezoid Midsegment Theorem and Subdivided Area Calculation
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Konular

Quadrilaterals and Polygons

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