High School

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In trapezoid CDEF, angle D = angle E = 90°, angle C = 60°, EF = 4.5 cm, and DE = 3.5 cm. What is the perimeter of the trapezoid to the nearest millimeter?

Describe your strategy.

(Note: The answer given in the textbook was 186 mm.)

Answer :

The perimeter of trapezoid CDEF is approximately 183 mm when calculated using the given dimensions and angles. This involves using trigonometry and the Pythagorean theorem to find missing side lengths. The textbook answer of 186 mm might result from different rounding or assumptions.

Step-by-Step Solution

  1. Since angles D and E are both 90°, CDEF is a right-angled trapezoid.

  2. We know DE = 3.5 cm and EF = 4.5 cm. Let's denote CD as a and CF as b.

  3. Notice that triangle CDE is a right triangle with angle C = 60°. We can use trigonometry to find CD and CF.

  4. Using the fact that tan(60°) = DE/CD, we get CD = DE/tan(60°) = 3.5/√3 = 2.02 cm (approximately).

  5. Next, we find CF using the Pythagorean theorem in triangle CDF.

  6. Since CD = 2.02 cm and DF = DE + EF = 3.5 + 4.5 = 8 cm:

    CF = √((CD)^2 + (DF)^2) = √((2.02)^2 + (8)^2) = √(4.08 + 64) = √68.08 ≈ 8.25 cm.

  7. Now we can calculate the perimeter of trapezoid CDEF:

    Perimeter = CD + DE + EF + CF ≈ 2.02 cm + 3.5 cm + 4.5 cm + 8.25 cm = 18.27 cm.

  8. Converting this to millimeters gives us 18.27 cm × 10 = 182.7 mm ≈ 183 mm (to the nearest millimeter).

  9. The perimeter of the trapezoid is approximately 183 mm.

Note: The textbook answer may be incorrect or based on different assumptions or rounding.

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