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Describe the altitude of a triangle.
The altitude extends from any vertex to the opposite side or to the line containing the opposite side at a 90° angle.
Compare right triangles and oblique triangles.
When can you use the Law of Sines to find a missing angle?
When the known values are the side opposite the missing angle and another side and its opposite angle.
In the Law of Sines, what is the relationship between the angle in the numerator and the side in the denominator?
What type of triangle results in an ambiguous case?
A triangle with two given sides and a non-included angle.
For the following exercises, assume $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Solve each triangle, if possible. Round each answer to the nearest tenth.
$\alpha =\mathrm{43\xb0},\gamma =\mathrm{69\xb0},a=20$
$\alpha =\mathrm{35\xb0},\gamma =\mathrm{73\xb0},c=20$
$\beta =\mathrm{72\xb0},a\approx 12.0,b\approx 19.9$
$\alpha =\mathrm{60\xb0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta =\mathrm{60\xb0},\text{\hspace{0.17em}}\gamma =\mathrm{60\xb0}$
$a=4,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\text{\hspace{0.17em}}\mathrm{60\xb0},\text{\hspace{0.17em}}\beta =\mathrm{100\xb0}$
$\gamma =\mathrm{20\xb0},b\approx 4.5,c\approx 1.6$
$b=10,\text{\hspace{0.17em}}\beta =\mathrm{95\xb0},\gamma =\text{\hspace{0.17em}}\mathrm{30\xb0}$
For the following exercises, use the Law of Sines to solve for the missing side for each oblique triangle. Round each answer to the nearest hundredth. Assume that angle $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ angle $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and angle $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.$
Find side $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}A=\mathrm{37\xb0},\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=\mathrm{49\xb0},\text{\hspace{0.17em}}c=5.$
$b\approx 3.78$
Find side $\text{\hspace{0.17em}}a$ when $\text{\hspace{0.17em}}A=\mathrm{132\xb0},C=\mathrm{23\xb0},b=10.$
Find side $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}B=\mathrm{37\xb0},C=21,\text{\hspace{0.17em}}b=23.$
$c\approx 13.70$
For the following exercises, assume $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}a,\beta \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}b,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\gamma \text{\hspace{0.17em}}$ is opposite side $\text{\hspace{0.17em}}c.\text{\hspace{0.17em}}$ Determine whether there is no triangle, one triangle, or two triangles. Then solve each triangle, if possible. Round each answer to the nearest tenth.
$\alpha =\mathrm{119\xb0},a=14,b=26$
$\gamma =\mathrm{113\xb0},b=10,c=32$
one triangle, $\text{\hspace{0.17em}}\alpha \approx \mathrm{50.3\xb0},\beta \approx \mathrm{16.7\xb0},a\approx 26.7$
$b=3.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=5.3,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\gamma =\text{\hspace{0.17em}}\mathrm{80\xb0}$
$a=12,\text{\hspace{0.17em}}\text{\hspace{0.17em}}c=17,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\text{\hspace{0.17em}}\mathrm{35\xb0}$
two triangles, $\text{\hspace{0.17em}}\gamma \approx \mathrm{54.3\xb0},\beta \approx \mathrm{90.7\xb0},b\approx 20.9$ or ${\gamma}^{\prime}\approx \mathrm{125.7\xb0},{\beta}^{\prime}\approx \mathrm{19.3\xb0},{b}^{\prime}\approx 6.9$
$a=20.5,\text{\hspace{0.17em}}\text{\hspace{0.17em}}b=35.0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta =\mathrm{25\xb0}$
$a=7,\text{\hspace{0.17em}}c=9,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =\text{\hspace{0.17em}}\mathrm{43\xb0}$
two triangles, $\beta \approx \mathrm{75.7\xb0},\gamma \approx \mathrm{61.3\xb0},b\approx 9.9$ or ${\beta}^{\prime}\approx \mathrm{18.3\xb0},{\gamma}^{\prime}\approx \mathrm{118.7\xb0},{b}^{\prime}\approx 3.2$
$a=7,b=3,\beta =\mathrm{24\xb0}$
$b=13,c=5,\gamma =\text{\hspace{0.17em}}\mathrm{10\xb0}$
two triangles, $\text{\hspace{0.17em}}\alpha \approx \mathrm{143.2\xb0},\beta \approx \mathrm{26.8\xb0},a\approx 17.3\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}{\alpha}^{\prime}\approx \mathrm{16.8\xb0},{\beta}^{\prime}\approx \mathrm{153.2\xb0},{a}^{\prime}\approx 8.3$
$a=2.3,c=1.8,\gamma =\mathrm{28\xb0}$
$\beta =\mathrm{119\xb0},b=8.2,a=11.3$
no triangle possible
For the following exercises, use the Law of Sines to solve, if possible, the missing side or angle for each triangle or triangles in the ambiguous case. Round each answer to the nearest tenth.
Find angle $A$ when $\text{\hspace{0.17em}}a=24,b=5,B=\mathrm{22\xb0.}$
Find angle $A$ when $\text{\hspace{0.17em}}a=13,b=6,B=\mathrm{20\xb0.}$
$A\approx \mathrm{47.8\xb0}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}{A}^{\prime}\approx \mathrm{132.2\xb0}$
Find angle $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}A=\mathrm{12\xb0},a=2,b=9.$
For the following exercises, find the area of the triangle with the given measurements. Round each answer to the nearest tenth.
$a=5,c=6,\beta =\text{\hspace{0.17em}}\mathrm{35\xb0}$
$8.6$
$b=11,c=8,\alpha =\mathrm{28\xb0}$
$a=32,b=24,\gamma =\mathrm{75\xb0}$
$370.9$
$a=7.2,b=4.5,\gamma =\mathrm{43\xb0}$
For the following exercises, find the length of side $\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ Round to the nearest tenth.
$12.3$
$12.2$
$16.0$
For the following exercises, find the measure of angle $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ if possible. Round to the nearest tenth.
$\mathrm{29.7\xb0}$
$x=\mathrm{76.9\xb0}\text{or}x=\mathrm{103.1\xb0}$
Notice that $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is an obtuse angle.
$\mathrm{110.6\xb0}$
For the following exercises, find the area of each triangle. Round each answer to the nearest tenth.
$A\approx 39.4,\text{}C\approx 47.6,\text{}BC\approx 20.7$
$57.1$
$42.0$
$430.2$
Find the radius of the circle in [link] . Round to the nearest tenth.
Find the diameter of the circle in [link] . Round to the nearest tenth.
$10.1$
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