Examkin

Question

Difficulty: Very hardPythagorean Theorem and Special Right Triangles

In △ABC\triangle ABC, the measure of ∠A\angle A is 45∘45^\circ, the measure of ∠C\angle C is 30∘30^\circ, and the length of side ACAC is 6+236 + 2\sqrt{3}. What is the length of side ABAB?

  1. 262\sqrt{6}Answer
  2. B
    626\sqrt{2}
  3. C
    434\sqrt{3}
  4. D
    232\sqrt{3}
  5. E
    32+63\sqrt{2} + \sqrt{6}

Answer

The length of side ABAB is 262\sqrt{6}.
The correct answer is 262\sqrt{6}. By drawing altitude BDBD from vertex BB to side ACAC, △ABC\triangle ABC is decomposed into right triangle BDABDA (a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle) and right triangle BDCBDC (a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle). Setting altitude BD=xBD = x gives AD=xAD = x and CD=x3CD = x\sqrt{3}. Combining these gives AC=x+x3=x(1+3)AC = x + x\sqrt{3} = x(1 + \sqrt{3}). Equating this to 6+23=23(1+3)6 + 2\sqrt{3} = 2\sqrt{3}(1 + \sqrt{3}) gives x=23x = 2\sqrt{3}. The hypotenuse ABAB of the 45∘−45∘−90∘45^\circ-45^\circ-90^\circ triangle is x2=(23)2=26x\sqrt{2} = (2\sqrt{3})\sqrt{2} = 2\sqrt{6}.

Step-by-Step Solution

1
Draw altitude BDBD perpendicular to side ACAC with point DD lying on segment ACAC.
△ABC\triangle ABC is partitioned into two adjacent right triangles: △BDA\triangle BDA and △BDC\triangle BDC.
Constructing an interior altitude allows the application of special right triangle ratio rules.
2
Analyze △BDC\triangle BDC (30∘−60∘−90∘30^\circ-60^\circ-90^\circ right triangle).
If BD=xBD = x, then CD=x3CD = x\sqrt{3} and BC=2xBC = 2x.
In a 30∘−60∘−90∘30^\circ-60^\circ-90^\circ triangle, sides opposite the angles are in the ratio 1:3:21 : \sqrt{3} : 2.
3
Analyze △BDA\triangle BDA (45∘−45∘−90∘45^\circ-45^\circ-90^\circ right triangle).
AD=BD=xAD = BD = x, and hypotenuse AB=x2AB = x\sqrt{2}.
In a 45∘−45∘−90∘45^\circ-45^\circ-90^\circ isosceles right triangle, legs are equal and the hypotenuse is leg×2\text{leg} \times \sqrt{2}.
4
Set up an equation for total side length AC=AD+CDAC = AD + CD.
x+x3=6+23  ⟹  x(1+3)=23(1+3)  ⟹  x=23x + x\sqrt{3} = 6 + 2\sqrt{3} \implies x(1 + \sqrt{3}) = 2\sqrt{3}(1 + \sqrt{3}) \implies x = 2\sqrt{3}.
Segment addition postulate combines ADAD and CDCD to match given total length ACAC.
5
Calculate requested side length ABAB.
AB=x2=(23)(2)=26AB = x\sqrt{2} = (2\sqrt{3})(\sqrt{2}) = 2\sqrt{6}.
Substitute x=23x = 2\sqrt{3} into the expression for hypotenuse ABAB.

Key Concept

Partitioning non-right triangles into 30∘−60∘−90∘30^\circ-60^\circ-90^\circ and 45∘−45∘−90∘45^\circ-45^\circ-90^\circ special right triangles by constructing an altitude.
Rate this question

Topics

Pythagorean Theorem and Special Right Triangles

More questions from this topic