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Difficulty: MediumQuadrilaterals and Polygons

In trapezoid ABCDABCD, side ABAB is parallel to side CDCD, and ∠DAB=90∘\angle DAB = 90^\circ. If AB=14AB = 14, CD=8CD = 8, and BC=10BC = 10, what is the area of trapezoid ABCDABCD?

  1. A
    40
  2. B
    66
  3. 88Answer
  4. D
    110
  5. E
    176

Answer

88
By drawing an altitude from point CC perpendicular to base ABAB at point HH, the trapezoid is partitioned into a rectangle ADCHADCH and a right triangle CHBCHB. The base of triangle CHBCHB is HB=14−8=6HB = 14 - 8 = 6, and its hypotenuse is BC=10BC = 10. Applying the Pythagorean theorem yields CH=102−62=8CH = \sqrt{10^2 - 6^2} = 8. Using the trapezoid area formula b1+b22×h\frac{b_1 + b_2}{2} \times h gives 14+82×8=88\frac{14 + 8}{2} \times 8 = 88.

Step-by-Step Solution

1
Decompose the trapezoid into a rectangle and a right triangle.
Draw a line segment from vertex CC perpendicular to base ABAB, intersecting ABAB at point HH. Quadrilateral ADCHADCH is a rectangle, and triangle CHBCHB is a right triangle with right angle at HH.
Splitting the trapezoid allows the perpendicular height to be determined using right triangle properties.
2
Determine the horizontal base of right triangle CHBCHB.
Since ADCHADCH is a rectangle, AH=CD=8AH = CD = 8. Therefore, HB=AB−AH=14−8=6HB = AB - AH = 14 - 8 = 6.
Opposite sides of a rectangle are equal in length.
3
Calculate height CHCH using the Pythagorean theorem.
In right triangle CHBCHB, CH2+HB2=BC2  ⟹  CH2+62=102  ⟹  CH2+36=100  ⟹  CH=64=8CH^2 + HB^2 = BC^2 \implies CH^2 + 6^2 = 10^2 \implies CH^2 + 36 = 100 \implies CH = \sqrt{64} = 8.
The height of the trapezoid corresponds to leg CHCH of right triangle CHBCHB.
4
Calculate the area of trapezoid ABCDABCD.
\text{Area} = \frac{AB + CD}{2} \times CH = \frac{14 + 8}{2} \times 8 = 11 \times 8 = 88.
The area formula for a trapezoid is half the sum of the parallel bases multiplied by the perpendicular height.

Key Concept

Decomposing a trapezoid into a rectangle and a right triangle to calculate height via the Pythagorean theorem
Estimated Time:1m 15s
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Topics

Quadrilaterals and Polygons

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