Examkin

Question

Difficulty: MediumPythagorean Theorem and Special Right Triangles

In the trapezoid ABCDABCD, bases ABAB and CDCD are parallel, and side ADAD is perpendicular to base ABAB. The length of ABAB is 1010 inches, the length of CDCD is 44 inches, and the measure of angle BB is 60∘60^\circ. What is the perimeter, in inches, of the trapezoid?

  1. A
    20+3320 + 3\sqrt{3}
  2. B
    20+6320 + 6\sqrt{3}
  3. 26+6326 + 6\sqrt{3}Answer
  4. D
    14+6314 + 6\sqrt{3}
  5. E
    32332\sqrt{3}

Answer

The perimeter of the trapezoid is 26+6326 + 6\sqrt{3} inches.
Drawing an altitude from vertex CC to base ABAB splits the trapezoid into a rectangle AECDAECD and a right triangle △CEB\triangle CEB. Since AE=CD=4AE = CD = 4, the leg EB=10−4=6EB = 10 - 4 = 6. The right triangle △CEB\triangle CEB is a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle where EBEB is the leg opposite the 30∘30^\circ angle. The hypotenuse BC=2×6=12BC = 2 \times 6 = 12 and the height CE=AD=63CE = AD = 6\sqrt{3}. Summing all four outer sides of the trapezoid (10+12+4+6310 + 12 + 4 + 6\sqrt{3}) yields the correct perimeter of 26+6326 + 6\sqrt{3} inches.

Step-by-Step Solution

1
Draw an altitude from vertex CC perpendicular to base ABAB at point EE.
A rectangle AECDAECD and a right triangle △CEB\triangle CEB are formed, with AE=CD=4AE = CD = 4 inches and CE=ADCE = AD.
To break down the trapezoid into a rectangle and a right triangle so we can find the unknown side lengths.
2
Find the length of segment EBEB.
EB=AB−AE=10−4=6EB = AB - AE = 10 - 4 = 6 inches.
To find the length of the base of the right triangle △CEB\triangle CEB.
3
Use the properties of a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle to determine the lengths of sides CECE (which is ADAD) and BCBC.
Since ∠B=60∘\angle B = 60^\circ is opposite to CECE, and EB=6EB = 6 is the shorter leg adjacent to 60∘60^\circ, the hypotenuse is BC=2×6=12BC = 2 \times 6 = 12 inches and the longer leg is CE=AD=63CE = AD = 6\sqrt{3} inches.
To compute the remaining unknown outer side lengths of the trapezoid.
4
Sum the four outer sides of the trapezoid to find the perimeter.
Perimeter = AB+BC+CD+DA=10+12+4+63=26+63AB + BC + CD + DA = 10 + 12 + 4 + 6\sqrt{3} = 26 + 6\sqrt{3} inches.
The perimeter is the total boundary length of the shape.

Key Concept

The perimeter of a right trapezoid can be found by drawing an altitude to create a rectangle and a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle, then determining the missing side lengths using special right triangle ratios.
Estimated Time:1m 30s
Rate this question

Topics

Pythagorean Theorem and Special Right Triangles

More questions from this topic