Question

for questions 11 - 14, solve the given triangles by finding the missing angle and other side lengths. (3 points each)
11.
triangle with vertices a, b, c; angle at a is 48°, angle at b is 75°, side bc is 7.8
12.
triangle with vertices a, b, (another vertex); angle at (one vertex) is 37°, angle at (another vertex) is 37°? wait, original ocr: angle 37°, side 200, angle 37°? wait, re - ocr: the second triangle has angle 37°, side 200, angle 37°? wait, the ocr text for the problem: for questions 11 - 14, solve the given triangles by finding the missing angle and other side lengths. (3 points each) then there are two triangle diagrams, one labeled 11 with angles 48° at a, 75° at b, side bc = 7.8; the other labeled 12 with some angles (37° and 37°?) and side 200. but the main ocr text is the problem statement and the two triangle images.

Explanation:

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Question 11 Steps:

Step1: Find missing angle $\angle C$

Sum of angles in triangle is $180^\circ$.
$\angle C = 180^\circ - 48^\circ - 75^\circ = 57^\circ$

Step2: Calculate side $AC$ via Law of Sines

Relate $\angle A$, $BC$, $\angle B$, $AC$.
$\frac{AC}{\sin 75^\circ} = \frac{7.8}{\sin 57^\circ}$
$AC = \frac{7.8 \times \sin 75^\circ}{\sin 57^\circ} \approx 9.9$

Step3: Calculate side $AB$ via Law of Sines

Relate $\angle C$, $AB$, $\angle B$, $AC$.
$\frac{AB}{\sin 57^\circ} = \frac{7.8}{\sin 57^\circ}$
$AB = \frac{7.8 \times \sin 48^\circ}{\sin 57^\circ} \approx 8.7$
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Question 12 Steps:

Step1: Find missing angle $\angle B$

Sum of angles in triangle is $180^\circ$.
$\angle B = 180^\circ - 37^\circ - 21^\circ = 122^\circ$

Step2: Calculate side $AB$ via Law of Sines

Relate $\angle C$, $AB$, $\angle B$, $AC$.
$\frac{AB}{\sin 21^\circ} = \frac{200}{\sin 122^\circ}$
$AB = \frac{200 \times \sin 21^\circ}{\sin 122^\circ} \approx 62.9$

Step3: Calculate side $BC$ via Law of Sines

Relate $\angle A$, $BC$, $\angle B$, $AC$.
$\frac{BC}{\sin 37^\circ} = \frac{200}{\sin 122^\circ}$
$BC = \frac{200 \times \sin 37^\circ}{\sin 122^\circ} \approx 249.2$

Answer:

Question 11:

$\angle C = 57^\circ$, $AC \approx 9.9$, $AB \approx 8.7$

Question 12:

$\angle B = 122^\circ$, $AB \approx 62.9$, $BC \approx 249.2$