All Circles Are Similar
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Pre-Calculus › All Circles Are Similar
A surveyor measures triangle sides $a = 80\text{ m}$, $b = 65\text{ m}$, and $c = 50\text{ m}$ (opposite angles $A$, $B$, and $C$ respectively). Using the measurements, which law should be applied first to find an angle of the triangle?
Use the Law of Sines because two angles and one side are known (AAS).
Use the Pythagorean theorem because the triangle must be right.
Use the Law of Cosines because three sides are known (SSS).
Use basic right-triangle tangent because an altitude is given.
Explanation
This problem asks which law to apply when all three sides of a triangle are known (SSS configuration). The Law of Cosines should be used first because it directly relates three sides and one angle: c² = a² + b² - 2ab cos(C). When you know all three sides (80 m, 65 m, and 50 m), you can rearrange to find any angle: cos(C) = (a² + b² - c²)/(2ab). The correct answer B properly identifies this as an SSS situation requiring the Law of Cosines. Option A incorrectly suggests using Law of Sines, which requires at least one angle to be known initially. Option C wrongly assumes the triangle is right-angled without verification, and option D incorrectly assumes an altitude is given. The strategy for SSS triangles is always to use the Law of Cosines first to find one angle, then you can use either law for the remaining angles.
In triangle $\triangle ABC$ formed during a construction layout, the measured sides are $AB=90\text{ m}$, $AC=70\text{ m}$, and $BC=50\text{ m}$. Using the measurements, which law should be applied first to solve for an angle of the triangle (such as $\angle A$)?
Use the Law of Cosines because three sides are known (SSS).
Use basic right-triangle trigonometry ($\tan$) because a height is implied.
Use the Law of Sines because two angles and one side are known (AAS).
Use the Pythagorean theorem because the triangle is right.
Explanation
This construction layout problem requires choosing the appropriate trigonometric law based on the given information. We should use the Law of Cosines because we know all three sides (90 m, 70 m, and 50 m), which is an SSS configuration. The Law of Cosines in the form cos(A) = (b² + c² - a²)/(2bc) allows us to find any angle when all three sides are known. The correct answer C properly identifies this as an SSS situation requiring the Law of Cosines. Option A incorrectly suggests Law of Sines for AAS (we don't have angles), option B wrongly assumes a right triangle, and option D incorrectly implies a height calculation. The key strategy is to identify what information is given (three sides = SSS), then select the appropriate law (Cosines for SSS/SAS, Sines for AAS/ASA).
A ship travels $40\text{ km}$ on bearing $045^\circ$, then travels $30\text{ km}$ on bearing $135^\circ$. Using the navigation measurements, how far is the ship from its starting point (in km)?
$50.0\text{ km}$
$70.0\text{ km}$
$10.0\text{ km}$
$56.6\text{ km}$
Explanation
This navigation problem requires the Law of Cosines to find the distance from start to finish. We use the Law of Cosines because we know two sides (40 km and 30 km) and can calculate the angle between them from the bearings, giving us an SAS configuration. The angle between the two paths is 135° - 45° = 90°, so the paths are perpendicular. Using the Law of Cosines: d² = 40² + 30² - 2(40)(30)cos(90°) = 1600 + 900 - 0 = 2500, so d = 50 km. The correct answer of 50 km makes sense because with perpendicular paths, we can also use the Pythagorean theorem directly. A common error would be to add the distances (70 km) or misinterpret the bearing angles. The key strategy is to find the angle between the two path segments from the bearings, recognize the SAS configuration, and apply the Law of Cosines.