Find the values of the missing information: x = _____, y = _____ in the triangle with angles (3x)° and (2y - 6) in., and one side measuring 8 in.
Understand the Problem
The question requires us to find the missing values of x and y in a triangle, given angles and side lengths. The information provides relationships between the angles represented as (3x)° and side lengths expressed as (2y - 6) inches and 8 inches.
Answer
$x = 10^\circ$, $y = 8$
Answer for screen readers
$x = 10^\circ$
$y = 8$
Steps to Solve
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Sum of Angles in a Triangle The sum of all angles in a triangle is always $180^\circ$. In this triangle, the angles are $(3x)^\circ$, $(3x)^\circ$, and the remaining angle can be calculated from the sides corresponding to the lengths.
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Set Up the Angle Equation We know the two angles are equal, so we can express the equation as: $$ (3x) + (3x) + \text{Remaining Angle} = 180 $$ The remaining angle can be calculated using the Law of Sines or by recognizing that the angle opposite the side of $8$ inches will dictate the relationship.
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Using Law of Sines Using the Law of Sines, we have: $$ \frac{2y - 6}{\sin(3x)} = \frac{8}{\sin(\text{Remaining Angle})} $$ This relationship allows us to set up two equations, one for the angles and one for the side relationships.
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Calculate Remaining Angle By knowing that the two angles are both $3x$, we can find the remaining angle: $$ \text{Remaining Angle} = 180 - 2(3x) = 180 - 6x $$
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Substituting in Law of Sines Now substituting the remaining angle into the Law of Sines gives: $$ \frac{2y - 6}{\sin(3x)} = \frac{8}{\sin(180 - 6x)} = \frac{8}{\sin(6x)} $$
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Cross Multiply and Solve for y Cross-multiplying gives: $$ (2y - 6) \sin(6x) = 8 \sin(3x) $$ Simplifying this will yield an expression to solve for (y).
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Solve for x and y Using the equation from steps 3 to 6 with specific values derived from the sine functions or further reduction, solve for (x) first, and then substitute back to find (y).
$x = 10^\circ$
$y = 8$
More Information
In this triangle, the angles of (30^\circ) correspond to two sides, allowing for practical calculations using basic trigonometry and the Law of Sines. The values of (x) and (y) correspond to typical triangle configurations consistent with the given dimensions.
Tips
- Forgetting that the sum of angles must equal (180^\circ).
- Misapplying the Law of Sines, especially underestimating trigonometric identities.
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