In right triangle ABC, the right angle is at vertex B, AB = 6, and the measure of \angle A… — GRE General Test

In right triangle $ABC$ , the right angle is at vertex $B$ , $AB = 6$ , and the measure of $\angle ACB$ is $30^\circ$ . Point $P$ lies on segment $BC$ such that the measure of $\angle APB$ is $60^\circ$ . Which of the following statements must be true? Select all that apply.

The length of segment $AP$ is $4\sqrt{3}$ .Answer
The length of segment $PC$ is $4\sqrt{3}$ .Answer
C
The area of triangle $APC$ is $18\sqrt{3}$ .
The perimeter of triangle $ABC$ is $18 + 6\sqrt{3}$ .Answer
E
The length of segment $BP$ is $3\sqrt{3}$ .

Answer

The correct statements are that the length of segment

AP

4\sqrt{3}

, the length of segment

PC

4\sqrt{3}

, and the perimeter of triangle

ABC

18 + 6\sqrt{3}

The 30-60-90 triangle ratio (

1 : \sqrt{3} : 2

) establishes

AC = 12

BC = 6\sqrt{3}

AP = 4\sqrt{3}

, and

BP = 2\sqrt{3}

. Consequently,

PC = 4\sqrt{3}

and the perimeter of triangle

ABC

equals

18 + 6\sqrt{3}

. Thus, the statements asserting that

AP = 4\sqrt{3}

PC = 4\sqrt{3}

, and the perimeter of triangle

ABC

18 + 6\sqrt{3}

are all correct.

Step-by-Step Solution

Analyze main triangle

ABC

Hypotenuse

AC = 12

and base

BC = 6\sqrt{3}

Triangle

ABC

is a 30-60-90 triangle with side opposite

30^\circ

equal to

AB = 6

. Therefore, hypotenuse

AC = 2(6) = 12

and leg

BC = 6\sqrt{3}

Analyze sub-triangle

ABP

Leg

BP = 2\sqrt{3}

and hypotenuse

AP = 4\sqrt{3}

Triangle

ABP

is a 30-60-90 triangle with side opposite

60^\circ

equal to

AB = 6

. The shorter leg is

BP = \frac{6}{\sqrt{3}} = 2\sqrt{3}

and the hypotenuse is

AP = 2(2\sqrt{3}) = 4\sqrt{3}

Calculate segment

PC

and area of triangle

APC

PC = 4\sqrt{3}

and

\text{Area}(\triangle APC) = 12\sqrt{3}

PC = BC - BP = 6\sqrt{3} - 2\sqrt{3} = 4\sqrt{3}

. The area of

\triangle APC

\frac{1}{2} \times PC \times AB = \frac{1}{2}(4\sqrt{3})(6) = 12\sqrt{3}

Calculate perimeter of triangle

ABC

Perimeter

= 18 + 6\sqrt{3}

Sum of sides

AB + BC + AC = 6 + 6\sqrt{3} + 12 = 18 + 6\sqrt{3}

Key Concept

Properties of 30-60-90 Special Right Triangles and Side Length Ratios (

1 : \sqrt{3} : 2

)

Question

Answer

Step-by-Step Solution

Key Concept

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