Examkin

Question

Difficulty: HardPythagorean Theorem and Special Right Triangles

In right triangle ABCABC, the measure of ∠B\angle B is 90∘90^\circ, and the measure of ∠C\angle C is 30∘30^\circ. Point DD lies on segment BCBC such that the measure of ∠ADC\angle ADC is 135∘135^\circ. If the length of segment ADAD is 88 units, what is the length, in units, of segment ACAC?

  1. A
    464\sqrt{6}
  2. B
    88
  3. 828\sqrt{2}Answer
  4. D
    1616
  5. E
    16216\sqrt{2}

Answer

828\sqrt{2}
The correct answer is 828\sqrt{2}. Since points BB, DD, and CC lie on a straight line, the adjacent angles ∠ADB\angle ADB and ∠ADC\angle ADC must sum to 180∘180^\circ. Subtracting the given angle measure shows that ∠ADB=180∘−135∘=45∘\angle ADB = 180^\circ - 135^\circ = 45^\circ. In the right triangle ABDABD, since ∠B=90∘\angle B = 90^\circ and ∠ADB=45∘\angle ADB = 45^\circ, the triangle is a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ special right triangle. Using the ratio of side lengths for this triangle type, the leg ABAB is equal to the hypotenuse ADAD divided by 2\sqrt{2}, which simplifies to AB=82=42AB = \frac{8}{\sqrt{2}} = 4\sqrt{2}. Next, looking at the larger right triangle ABCABC, the angle at CC is given as 30∘30^\circ, which makes △ABC\triangle ABC a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ special right triangle. In this type of triangle, the hypotenuse ACAC is twice the length of the shorter leg ABAB, which is opposite the 30∘30^\circ angle. Multiplying the length of ABAB by 22 yields AC=2×42=82AC = 2 \times 4\sqrt{2} = 8\sqrt{2} units.

Step-by-Step Solution

1
Find the measure of angle ADBADB using the supplementary angle relationship along the line segment BCBC.
∠ADB=180∘−135∘=45∘\angle ADB = 180^\circ - 135^\circ = 45^\circ
Points BB, DD, and CC are collinear, meaning ∠ADB\angle ADB and ∠ADC\angle ADC form a linear pair and must sum to 180∘180^\circ.
2
Determine the properties of right triangle ABDABD and solve for the length of side ABAB.
△ABD\triangle ABD is a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle, where leg AB=AD2=82=42AB = \frac{AD}{\sqrt{2}} = \frac{8}{\sqrt{2}} = 4\sqrt{2} units.
Since ∠B=90∘\angle B = 90^\circ and ∠ADB=45∘\angle ADB = 45^\circ, the remaining angle ∠DAB\angle DAB is also 45∘45^\circ. In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle, the leg length equals the hypotenuse divided by 2\sqrt{2}.
3
Use the properties of the larger 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle ABCABC to find the length of hypotenuse ACAC.
AC=2×AB=2×42=82AC = 2 \times AB = 2 \times 4\sqrt{2} = 8\sqrt{2} units.
In right triangle ABCABC, the angle opposite leg ABAB is ∠C=30∘\angle C = 30^\circ. In any 30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle, the hypotenuse is exactly twice the length of the leg opposite the 30∘30^\circ angle.

Key Concept

Using multi-step properties of special right triangles (45∘−45∘−90∘45^\circ-45^\circ-90^\circ and 30∘−60∘−90∘30^\circ-60^\circ-90^\circ) sharing a common boundary line.
Rate this question

Topics

Pythagorean Theorem and Special Right Triangles

More questions from this topic