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Difficulty: MediumPythagorean Theorem and Special Right Triangles

A park planner is designing a triangular walking path ABCABC where the corner at vertex BB forms a 90∘90^\circ angle. A straight path ADAD is constructed from vertex AA to point DD on side BCBC, dividing angle ∠BAC\angle BAC into two equal angles measuring 30∘30^\circ each. If side AB=18AB = 18 meters, what is the length, in meters, of segment DCDC?

  1. A
    636\sqrt{3}
  2. B
    939\sqrt{3}
  3. 12312\sqrt{3}Answer
  4. D
    18318\sqrt{3}
  5. E
    3636

Answer

The length of segment DCDC is 12312\sqrt{3} meters.
In right triangle ABDABD, ∠BAD=30∘\angle BAD = 30^\circ and AB=18AB = 18, so BD=183=63BD = \frac{18}{\sqrt{3}} = 6\sqrt{3} meters. In right triangle ABCABC, ∠BAC=60∘\angle BAC = 60^\circ, so BC=183BC = 18\sqrt{3} meters. Subtracting BDBD from BCBC gives DC=183−63=123DC = 18\sqrt{3} - 6\sqrt{3} = 12\sqrt{3} meters.

Step-by-Step Solution

1
Determine the angles in right triangle ABDABD and right triangle ABCABC.
In △ABD\triangle ABD, ∠B=90∘\angle B = 90^\circ and ∠BAD=30∘\angle BAD = 30^\circ. In △ABC\triangle ABC, ∠B=90∘\angle B = 90^\circ and ∠BAC=30∘+30∘=60∘\angle BAC = 30^\circ + 30^\circ = 60^\circ.
Path ADAD bisects ∠BAC\angle BAC into two 30∘30^\circ angles.
2
Calculate the length of segment BDBD using the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle ratio in △ABD\triangle ABD.
BD=AB3=183=63BD = \frac{AB}{\sqrt{3}} = \frac{18}{\sqrt{3}} = 6\sqrt{3} meters.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, the leg opposite the 30∘30^\circ angle is equal to the adjacent leg divided by 3\sqrt{3}.
3
Calculate the total length of leg BCBC using the 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle ratio in △ABC\triangle ABC.
BC=AB⋅3=183BC = AB \cdot \sqrt{3} = 18\sqrt{3} meters.
In △ABC\triangle ABC, the leg opposite the 60∘60^\circ angle (BCBC) is 3\sqrt{3} times the adjacent leg (AB=18AB = 18).
4
Subtract segment BDBD from total leg BCBC to find segment DCDC.
DC=BC−BD=183−63=123DC = BC - BD = 18\sqrt{3} - 6\sqrt{3} = 12\sqrt{3} meters.
Segment addition postulate state that BD+DC=BCBD + DC = BC.

Key Concept

Properties of 30∘−60∘−90∘30^\circ-60^\circ-90^\circ Special Right Triangles
Estimated Time:1m 30s
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Topics

Pythagorean Theorem and Special Right Triangles

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