two sides and an angle (ssa) of a triangle are given. determine whether the given measurements produce one triangle, two triangles, or no triangle at all. solve each triangle that results.

$a = 15$, $c = 6.1$, $a = 62^\circ$

selected the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (round side lengths to the nearest tenth and angle measurements to the nearest degree as needed.)

a. there is only one possible solution for the triangle.
the measurements for the remaining side $b$ and angles $c$ and $b$ are as follows.
$c \approx \box^\circ$
$b \approx \box^\circ$
$b \approx \box$

b. there are two possible solutions for the triangle.
the measurements for the solution with the smaller angle $c$ are as follows.
$c_1 \approx \box^\circ$
$b_1 \approx \box^\circ$
$b_1 \approx \box$
the measurements for the solution with the larger angle $c$ are as follows.
$c_2 \approx \box^\circ$
$b_2 \approx \box^\circ$
$b_2 \approx \box$

c. there are no possible solutions for this triangle.

Question

two sides and an angle (ssa) of a triangle are given. determine whether the given measurements produce one triangle, two triangles, or no triangle at all. solve each triangle that results.

$a = 15$, $c = 6.1$, $a = 62^\circ$

selected the correct choice below and, if necessary, fill in the answer boxes to complete your choice. (round side lengths to the nearest tenth and angle measurements to the nearest degree as needed.)

a. there is only one possible solution for the triangle.
the measurements for the remaining side $b$ and angles $c$ and $b$ are as follows.
$c \approx \box^\circ$
$b \approx \box^\circ$
$b \approx \box$

b. there are two possible solutions for the triangle.
the measurements for the solution with the smaller angle $c$ are as follows.
$c_1 \approx \box^\circ$
$b_1 \approx \box^\circ$
$b_1 \approx \box$
the measurements for the solution with the larger angle $c$ are as follows.
$c_2 \approx \box^\circ$
$b_2 \approx \box^\circ$
$b_2 \approx \box$

c. there are no possible solutions for this triangle.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Law of Sines", "Solving Triangles" ], "new_concepts": [], "current_concepts": [ "Law of Sines", "Solving Triangles", "SSA Ambiguous Case" ] } ## Apply the Law of Sines to find angle C $$ \begin{aligned} \frac{\sin(C)}{c} &= \frac{\sin(A)}{a} \ \sin(C) &= \frac{c \cdot \sin(A)}{a} \ \sin(C) &= \frac{6.1 \cdot \sin(62^\circ)}{15} \ \sin(C) &\approx \frac{6.1 \cdot 0.88295}{15} \approx 0.35907 \ C_1 &\approx \arcsin(0.35907) \approx 21.04^\circ \approx 21^\circ \ C_2 &= 180^\circ - C_1 \approx 158.96^\circ \approx 159^\circ \end{aligned} $$ ## Check the validity of the two cases $$ \begin{aligned} \text{Case 1: } C_1 &\approx 21^\circ \ B_1 &= 180^\circ - A - C_1 = 180^\circ - 62^\circ - 21^\circ = 97^\circ \quad (\text{Valid}) \ \text{Case 2: } C_2 &\approx 159^\circ \ B_2 &= 180^\circ - A - C_2 = 180^\circ - 62^\circ - 159^\circ = -41^\circ \quad (\text{Invalid}) \end{aligned} $$ ## Solve for the remaining side b in the valid case $$ \begin{aligned} C &\approx 21^\circ, \quad B \approx 97^\circ \ \frac{b}{\sin(B)} &= \frac{a}{\sin(A)} \ b &= \frac{15 \cdot \sin(97^\circ)}{\sin(62^\circ)} \ b &\approx \frac{15 \cdot 0.99255}{0.88295} \approx 16.86 \approx 16.9 \end{aligned} $$

A. There is only one possible solution for the triangle. The measurements for the remaining side b and angles C and B are as follows. $C \approx 21^\circ$ $B \approx 97^\circ$ $b \approx 16.9$

B. There are two possible solutions for the triangle. The measurements for the solution with the smaller angle C are as follows. $C_1 \approx \text{ }$ $B_1 \approx \text{ }$ $b_1 \approx \text{ }$ The measurements for the solution with the larger angle C are as follows. $C_2 \approx \text{ }$ $B_2 \approx \text{ }$ $b_2 \approx \text{ }$

C. There are no possible solutions for this triangle.

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Law of Sines" ] }

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QUESTION IMAGE

Question

Explanation:

Apply the Law of Sines to find angle C

Check the validity of the two cases

Solve for the remaining side b in the valid case

Answer: