All Circles Are Similar - Geometry
Card 1 of 30
Identify the special simplification of Law of Cosines when the included angle is $90^\circ$.
Identify the special simplification of Law of Cosines when the included angle is $90^\circ$.
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$a^2=b^2+c^2$ when $A=90^\circ$. Reduces to Pythagorean theorem when angle is right.
$a^2=b^2+c^2$ when $A=90^\circ$. Reduces to Pythagorean theorem when angle is right.
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Find the resultant magnitude $R$ of two forces $10$ and $6$ with included angle $60^\circ$.
Find the resultant magnitude $R$ of two forces $10$ and $6$ with included angle $60^\circ$.
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$R=\sqrt{10^2+6^2+2\cdot10\cdot^6\cos 60^\circ}=\sqrt{196}=14$. Vector addition using cosine formula for resultant.
$R=\sqrt{10^2+6^2+2\cdot10\cdot^6\cos 60^\circ}=\sqrt{196}=14$. Vector addition using cosine formula for resultant.
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Identify the key calculator mode needed when using $\sin^{-1}$ or $\cos^{-1}$ with degrees.
Identify the key calculator mode needed when using $\sin^{-1}$ or $\cos^{-1}$ with degrees.
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Degree mode. Calculator must be set for degree calculations.
Degree mode. Calculator must be set for degree calculations.
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Find $C$ (nearest degree) if $A=40^\circ$, $a=12$, and $c=9$ using Law of Sines.
Find $C$ (nearest degree) if $A=40^\circ$, $a=12$, and $c=9$ using Law of Sines.
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$C\approx\sin^{-1}!\left(\frac{9\sin 40^\circ}{12}\right)\approx 29^\circ$. Use inverse sine with the established proportion.
$C\approx\sin^{-1}!\left(\frac{9\sin 40^\circ}{12}\right)\approx 29^\circ$. Use inverse sine with the established proportion.
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Find $b$ if $a=18$, $A=50^\circ$, and $B=80^\circ$ (nearest tenth).
Find $b$ if $a=18$, $A=50^\circ$, and $B=80^\circ$ (nearest tenth).
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$b\approx 18\frac{\sin 80^\circ}{\sin 50^\circ}\approx 23.1$. Cross-multiply the sine proportion to solve.
$b\approx 18\frac{\sin 80^\circ}{\sin 50^\circ}\approx 23.1$. Cross-multiply the sine proportion to solve.
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What condition guarantees exactly one right triangle in the $SSA$ case when $A$ is acute and $a$ is given?
What condition guarantees exactly one right triangle in the $SSA$ case when $A$ is acute and $a$ is given?
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$a=h$ where $h=c\sin A$. Side exactly equals height, forming right triangle.
$a=h$ where $h=c\sin A$. Side exactly equals height, forming right triangle.
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Identify whether $a=6$, $c=12$, $A=30^\circ$ gives $0$, $1$, or $2$ triangles using $h=c\sin A$.
Identify whether $a=6$, $c=12$, $A=30^\circ$ gives $0$, $1$, or $2$ triangles using $h=c\sin A$.
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$1$ triangle (right) because $h=6$ and $a=h$. Side equals height, creating perpendicular configuration.
$1$ triangle (right) because $h=6$ and $a=h$. Side equals height, creating perpendicular configuration.
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Identify whether $a=14$, $c=12$, $A=30^\circ$ gives $0$, $1$, or $2$ triangles.
Identify whether $a=14$, $c=12$, $A=30^\circ$ gives $0$, $1$, or $2$ triangles.
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$1$ triangle because $a\ge c$. Opposite side is longer than adjacent side.
$1$ triangle because $a\ge c$. Opposite side is longer than adjacent side.
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Find $c$ if $a=13$, $b=14$, and $C=60^\circ$ using the Law of Cosines (nearest tenth).
Find $c$ if $a=13$, $b=14$, and $C=60^\circ$ using the Law of Cosines (nearest tenth).
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$c\approx\sqrt{13^2+14^2-2\cdot13\cdot14\cos 60^\circ}\approx 13.5$. Apply cosine formula with two sides and included angle.
$c\approx\sqrt{13^2+14^2-2\cdot13\cdot14\cos 60^\circ}\approx 13.5$. Apply cosine formula with two sides and included angle.
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Find $A$ if $a=8$, $b=5$, and $c=7$ using $\cos A$ (nearest degree).
Find $A$ if $a=8$, $b=5$, and $c=7$ using $\cos A$ (nearest degree).
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$A\approx\cos^{-1}!\left(\frac{5^2+7^2-8^2}{2\cdot^5\cdot^7}\right)\approx 79^\circ$. Use inverse cosine formula for finding angles.
$A\approx\cos^{-1}!\left(\frac{5^2+7^2-8^2}{2\cdot^5\cdot^7}\right)\approx 79^\circ$. Use inverse cosine formula for finding angles.
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What condition guarantees no triangle in the $SSA$ case when given $A$ (acute), $a$, and adjacent side $c$?
What condition guarantees no triangle in the $SSA$ case when given $A$ (acute), $a$, and adjacent side $c$?
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$a<h$ where $h=c\sin A$. Side opposite acute angle is too short to reach.
$a<h$ where $h=c\sin A$. Side opposite acute angle is too short to reach.
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Find $B$ if $b=15$, $a=10$, and $c=12$ using $\cos B$ (nearest degree).
Find $B$ if $b=15$, $a=10$, and $c=12$ using $\cos B$ (nearest degree).
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$B\approx\cos^{-1}!\left(\frac{10^2+12^2-15^2}{2\cdot10\cdot12}\right)\approx 104^\circ$. Apply cosine angle formula with three known sides.
$B\approx\cos^{-1}!\left(\frac{10^2+12^2-15^2}{2\cdot10\cdot12}\right)\approx 104^\circ$. Apply cosine angle formula with three known sides.
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Identify the correct proportion to solve for $b$ if $a$, $A$, $B$ are known using Law of Sines.
Identify the correct proportion to solve for $b$ if $a$, $A$, $B$ are known using Law of Sines.
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$\frac{b}{\sin B}=\frac{a}{\sin A}$. Standard proportion setup for unknown side calculation.
$\frac{b}{\sin B}=\frac{a}{\sin A}$. Standard proportion setup for unknown side calculation.
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Find the third side $c$ of a triangle if $a=9$, $b=9$, and $C=60^\circ$.
Find the third side $c$ of a triangle if $a=9$, $b=9$, and $C=60^\circ$.
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$c=\sqrt{9^2+9^2-2\cdot^9\cdot^9\cos 60^\circ}=9$. Isosceles triangle with 60° creates equilateral result.
$c=\sqrt{9^2+9^2-2\cdot^9\cdot^9\cos 60^\circ}=9$. Isosceles triangle with 60° creates equilateral result.
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What is the triangle angle-sum equation you must use after finding two angles with trig laws?
What is the triangle angle-sum equation you must use after finding two angles with trig laws?
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$A+B+C=180^\circ$. All triangle angles must sum to 180 degrees.
$A+B+C=180^\circ$. All triangle angles must sum to 180 degrees.
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Identify whether Law of Sines or Law of Cosines is best first for given $A$, $B$, and side $a$.
Identify whether Law of Sines or Law of Cosines is best first for given $A$, $B$, and side $a$.
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Law of Sines. Two angles and one side use sine relationships.
Law of Sines. Two angles and one side use sine relationships.
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Identify whether Law of Sines or Law of Cosines is best first for given $a$, $b$, and included angle $C$.
Identify whether Law of Sines or Law of Cosines is best first for given $a$, $b$, and included angle $C$.
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Law of Cosines. Two sides with included angle need cosine approach.
Law of Cosines. Two sides with included angle need cosine approach.
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Find $C$ if $A=35^\circ$ and $B=80^\circ$.
Find $C$ if $A=35^\circ$ and $B=80^\circ$.
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$C=180^\circ-35^\circ-80^\circ=65^\circ$. Apply angle sum property of triangles.
$C=180^\circ-35^\circ-80^\circ=65^\circ$. Apply angle sum property of triangles.
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Identify the second possible angle $B$ in the $SSA$ case if $\sin B=0.5$ and $B$ is not acute.
Identify the second possible angle $B$ in the $SSA$ case if $\sin B=0.5$ and $B$ is not acute.
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$B=180^\circ-30^\circ=150^\circ$. Supplementary angle when sine has two solutions.
$B=180^\circ-30^\circ=150^\circ$. Supplementary angle when sine has two solutions.
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Find $B$ if $a=12$, $A=40^\circ$, and $b=9$ using the Law of Sines (nearest degree).
Find $B$ if $a=12$, $A=40^\circ$, and $b=9$ using the Law of Sines (nearest degree).
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$B\approx\sin^{-1}!\left(\frac{9\sin 40^\circ}{12}\right)\approx 29^\circ$. Use inverse sine after setting up the proportion.
$B\approx\sin^{-1}!\left(\frac{9\sin 40^\circ}{12}\right)\approx 29^\circ$. Use inverse sine after setting up the proportion.
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Find $c$ using the Law of Sines if $a=8$, $A=60^\circ$, and $C=45^\circ$.
Find $c$ using the Law of Sines if $a=8$, $A=60^\circ$, and $C=45^\circ$.
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$c=8\frac{\sin 45^\circ}{\sin 60^\circ}=\frac{8\sqrt{6}}{3}$. Set up proportion and solve for unknown side.
$c=8\frac{\sin 45^\circ}{\sin 60^\circ}=\frac{8\sqrt{6}}{3}$. Set up proportion and solve for unknown side.
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Find $a$ using the Law of Sines if $b=10$, $B=30^\circ$, and $A=45^\circ$.
Find $a$ using the Law of Sines if $b=10$, $B=30^\circ$, and $A=45^\circ$.
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$a=10\frac{\sin 45^\circ}{\sin 30^\circ}=10\sqrt{2}$. Cross-multiply and substitute known angle values.
$a=10\frac{\sin 45^\circ}{\sin 30^\circ}=10\sqrt{2}$. Cross-multiply and substitute known angle values.
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What is the common labeling convention relating $a$ and $A$ in triangle $ABC$?
What is the common labeling convention relating $a$ and $A$ in triangle $ABC$?
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$a$ is opposite $A$. Standard notation: lowercase sides opposite uppercase angles.
$a$ is opposite $A$. Standard notation: lowercase sides opposite uppercase angles.
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Which method is most appropriate first for an $SAS$ triangle: Law of Sines or Law of Cosines?
Which method is most appropriate first for an $SAS$ triangle: Law of Sines or Law of Cosines?
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Law of Cosines. Two sides and included angle need the cosine formula.
Law of Cosines. Two sides and included angle need the cosine formula.
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What condition yields no triangle in the $SSA$ case when the given angle $A$ is obtuse?
What condition yields no triangle in the $SSA$ case when the given angle $A$ is obtuse?
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$a\le c$. Obtuse angle cannot have shorter opposite side.
$a\le c$. Obtuse angle cannot have shorter opposite side.
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What condition yields exactly one triangle in the $SSA$ case when the given angle $A$ is obtuse?
What condition yields exactly one triangle in the $SSA$ case when the given angle $A$ is obtuse?
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$a>c$. Obtuse angle requires opposite side to be longest.
$a>c$. Obtuse angle requires opposite side to be longest.
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What condition yields exactly one triangle in the $SSA$ case when $A$ is acute and the opposite side is long enough?
What condition yields exactly one triangle in the $SSA$ case when $A$ is acute and the opposite side is long enough?
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$a\ge c$. Opposite side is longer than adjacent, ensuring uniqueness.
$a\ge c$. Opposite side is longer than adjacent, ensuring uniqueness.
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What condition yields two possible triangles in the $SSA$ case when $A$ is acute and adjacent side is $c$?
What condition yields two possible triangles in the $SSA$ case when $A$ is acute and adjacent side is $c$?
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$h<a<c$ where $h=c\sin A$. Side length allows two possible triangle configurations.
$h<a<c$ where $h=c\sin A$. Side length allows two possible triangle configurations.
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State the Law of Cosines form typically used for resultant vectors with included angle $\theta$.
State the Law of Cosines form typically used for resultant vectors with included angle $\theta$.
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$R^2=p^2+q^2+2pq\cos\theta$. General form for vector resultant calculations.
$R^2=p^2+q^2+2pq\cos\theta$. General form for vector resultant calculations.
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State the Law of Sines for triangle $ABC$ with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$.
State the Law of Sines for triangle $ABC$ with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$.
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$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates ratios of sides to sines of opposite angles.
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. Relates ratios of sides to sines of opposite angles.
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