Question

State whether the given measurements determine zero, one, or two triangles.

A = 61°, a = 23, b = 24

Help me please

Answer 1

Answer on Question #42403 – Math - Geometry

Task

State whether the given measurements determine zero, one, or two triangles.

$$A = 61{}^\circ, \quad a = 23, \quad b = 24$$

Solution

The law of cosines (also known as the cosine formula or cosine law)

$$a^2 = b^2 + c^2 - 2bc \cos A$$ $$23^2 = 24^2 + c^2 - 2 * 24 * c \cos 61{}^\circ$$ $$c^2 - 48 * 0.485 * c + 576 - 529 = 0$$ $$c^2 - 23.28c + 47 = 0$$ $$c = \frac{23.28 \pm \sqrt{23.28^2 - 4 * 47}}{2}$$ $$c = \frac{23.28 \pm 18.81}{2}$$ $$c = 21.045 \text{ or } c = 2.235$$

1) $c = 21.045$

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ $$B = \sin^{-1} \frac{b \sin A}{a} = \sin^{-1} \frac{24 * 0.875}{23} = \sin^{-1} 0.91304 \dots = 65.93{}^\circ$$ $$C = 180{}^\circ - A - B = 180{}^\circ - 61{}^\circ - 65.93{}^\circ = 53.07{}^\circ$$

2) $c = 2.235$

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ $$\sin B = \sin(\pi - B) = \sin(180{}^\circ - B)$$ $$180{}^\circ - B = \sin^{-1} \frac{24 * 0.875}{23} = \sin^{-1} 0.91304 \dots = 65.93{}^\circ$$ $$B = 180{}^\circ - 65.93{}^\circ = 114.07{}^\circ$$ $$C = 180{}^\circ - A - B = 180{}^\circ - 61{}^\circ - 114.07{}^\circ = 4.93{}^\circ$$

Answer:

1) $A = 61{}^\circ$ $\quad a = 23,$

$$B = 65.93{}^\circ \quad \quad b = 24$$ $$C = 53.07{}^\circ \quad \quad c = 21.045$$

2) $A = 61{}^\circ$ $\quad a = 23,$

$$B = 114.07{}^{\circ} \quad b = 24$$ $$C = 4.93{}^{\circ} \quad c = 2.235$$

There are 2 triangles.

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