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Difficulty: Very hardPythagorean Theorem and Special Right Triangles

In the coordinate plane, segment ABAB lies along the positive xx-axis with point AA at the origin (0,0)(0,0) and point BB at (63,0)(6\sqrt{3}, 0). Point CC is located in the first quadrant such that △ABC\triangle ABC is a right triangle with ∠ACB=90∘\angle ACB = 90^\circ and ∠CAB=30∘\angle CAB = 30^\circ. Point DD is also located in the first quadrant such that △ABD\triangle ABD is an isosceles right triangle with hypotenuse ABAB and ∠ADB=90∘\angle ADB = 90^\circ.

Which of the following statements regarding this figure are true? Select all that apply.

  1. The length of segment ACAC is 99.Answer
  2. The length of segment ADAD is 363\sqrt{6}.Answer
  3. The area of triangle ABDABD is 2727.Answer
  4. D
    The length of segment BCBC is 99.
  5. E
    The length of segment ADAD is 666\sqrt{6}.

Answer

The correct statements are: the length of segment ACAC is 99, the length of segment ADAD is 363\sqrt{6}, and the area of triangle ABDABD is 2727.
The statement that AC=9AC = 9 is correct because in △ABC\triangle ABC, AC=ABcos⁡(30∘)=63⋅32=9AC = AB \cos(30^\circ) = 6\sqrt{3} \cdot \frac{\sqrt{3}}{2} = 9. The statement that AD=36AD = 3\sqrt{6} is correct because in isosceles right △ABD\triangle ABD, AD=AB2=632=36AD = \frac{AB}{\sqrt{2}} = \frac{6\sqrt{3}}{\sqrt{2}} = 3\sqrt{6}. The statement that the area of △ABD\triangle ABD is 2727 is correct because 12(36)(36)=27\frac{1}{2}(3\sqrt{6})(3\sqrt{6}) = 27.

Step-by-Step Solution

1
Determine the side lengths of 30-60-90 triangle ABC
Hypotenuse AB=63AB = 6\sqrt{3}. Leg BCBC opposite 30∘30^\circ is 12(63)=33\frac{1}{2}(6\sqrt{3}) = 3\sqrt{3}. Leg ACAC opposite 60∘60^\circ is 33⋅3=93\sqrt{3} \cdot \sqrt{3} = 9.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, side ratios are 1:3:21 : \sqrt{3} : 2 relative to angles 30∘:60∘:90∘30^\circ : 60^\circ : 90^\circ.
2
Determine the side lengths of 45-45-90 triangle ABD
Hypotenuse AB=63AB = 6\sqrt{3}. Legs AD=BD=632=36AD = BD = \frac{6\sqrt{3}}{\sqrt{2}} = 3\sqrt{6}.
In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ isosceles right triangle, side ratios are 1:1:21 : 1 : \sqrt{2}, so leg length equals hypotenuse divided by 2\sqrt{2}.
3
Calculate the area of right triangle ABD
Area (△ABD)=12⋅AD⋅BD=12(36)(36)=12(54)=27(\triangle ABD) = \frac{1}{2} \cdot AD \cdot BD = \frac{1}{2} (3\sqrt{6})(3\sqrt{6}) = \frac{1}{2} (54) = 27.
The area of a right triangle is half the product of its perpendicular legs.
4
Evaluate each given option against computed values
Segment AC=9AC = 9 is true. Segment AD=36AD = 3\sqrt{6} is true. Area of △ABD=27\triangle ABD = 27 is true. Segment BC=9BC = 9 is false (BC=33BC = 3\sqrt{3}). Segment AD=66AD = 6\sqrt{6} is false (AD=36AD = 3\sqrt{6}).
Direct comparison with calculated geometric dimensions.

Key Concept

Side ratios of 30-60-90 (1:3:21:\sqrt{3}:2) and 45-45-90 (1:1:21:1:\sqrt{2}) special right triangles
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Topics

Pythagorean Theorem and Special Right Triangles

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