Westonci.ca is the Q&A platform that connects you with experts who provide accurate and detailed answers. Get quick and reliable solutions to your questions from a community of seasoned experts on our user-friendly platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To find the angles in triangle [tex]\(ABC\)[/tex] where [tex]\(32A = 42B = 62C\)[/tex], follow these steps:
1. Set Up the Proportionality:
We need to introduce a common constant [tex]\( k \)[/tex] such that:
[tex]\[ 32A = 42B = 62C = k \][/tex]
2. Express Each Angle in Terms of [tex]\( k \)[/tex]:
[tex]\[ A = \frac{k}{32} \][/tex]
[tex]\[ B = \frac{k}{42} \][/tex]
[tex]\[ C = \frac{k}{62} \][/tex]
3. Utilize the Sum of Angles in a Triangle:
The sum of the angles in any triangle is [tex]\( 180^\circ \)[/tex]. Therefore,
[tex]\[ A + B + C = 180^\circ \][/tex]
Substitute the expressions in terms of [tex]\( k \)[/tex]:
[tex]\[ \frac{k}{32} + \frac{k}{42} + \frac{k}{62} = 180^\circ \][/tex]
4. Find a Common Denominator and Combine:
The least common multiple of 32, 42, and 62 is 249984:
[tex]\[ \frac{249984}{32} = 7812, \quad \frac{249984}{42} = 5952, \quad \frac{249984}{62} = 4032 \][/tex]
Now rewrite the equation using the common denominator:
[tex]\[ \frac{k \cdot 7812 + k \cdot 5952 + k \cdot 4032}{249984} = 180 \][/tex]
5. Sum the Numerators and Simplify:
[tex]\[ \frac{k (7812 + 5952 + 4032)}{249984} = 180 \][/tex]
[tex]\[ \frac{k \cdot 17796}{249984} = 180 \][/tex]
Simplify the fraction:
[tex]\[ \frac{k}{14.04} = 180 \][/tex]
Multiply both sides by 14.04:
[tex]\[ k = 2510.4 \times 180 = 451872 \][/tex]
6. Solve for Each Angle:
Now substitute [tex]\( k \)[/tex] back into the expressions for angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
[tex]\[ A = \frac{451872}{32} \approx 14121 \quad B = \frac{451872}{42} \approx 10759.33 \quad C = \frac{451872}{62} \approx 7288.26 \][/tex]
Therefore, the angles in triangle [tex]\(ABC\)[/tex] are approximately:
[tex]\[ A \approx 79^\circ.00, \quad B \approx 60^\circ.23, \quad C \approx 40^\circ.80 \][/tex]
1. Set Up the Proportionality:
We need to introduce a common constant [tex]\( k \)[/tex] such that:
[tex]\[ 32A = 42B = 62C = k \][/tex]
2. Express Each Angle in Terms of [tex]\( k \)[/tex]:
[tex]\[ A = \frac{k}{32} \][/tex]
[tex]\[ B = \frac{k}{42} \][/tex]
[tex]\[ C = \frac{k}{62} \][/tex]
3. Utilize the Sum of Angles in a Triangle:
The sum of the angles in any triangle is [tex]\( 180^\circ \)[/tex]. Therefore,
[tex]\[ A + B + C = 180^\circ \][/tex]
Substitute the expressions in terms of [tex]\( k \)[/tex]:
[tex]\[ \frac{k}{32} + \frac{k}{42} + \frac{k}{62} = 180^\circ \][/tex]
4. Find a Common Denominator and Combine:
The least common multiple of 32, 42, and 62 is 249984:
[tex]\[ \frac{249984}{32} = 7812, \quad \frac{249984}{42} = 5952, \quad \frac{249984}{62} = 4032 \][/tex]
Now rewrite the equation using the common denominator:
[tex]\[ \frac{k \cdot 7812 + k \cdot 5952 + k \cdot 4032}{249984} = 180 \][/tex]
5. Sum the Numerators and Simplify:
[tex]\[ \frac{k (7812 + 5952 + 4032)}{249984} = 180 \][/tex]
[tex]\[ \frac{k \cdot 17796}{249984} = 180 \][/tex]
Simplify the fraction:
[tex]\[ \frac{k}{14.04} = 180 \][/tex]
Multiply both sides by 14.04:
[tex]\[ k = 2510.4 \times 180 = 451872 \][/tex]
6. Solve for Each Angle:
Now substitute [tex]\( k \)[/tex] back into the expressions for angles [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]:
[tex]\[ A = \frac{451872}{32} \approx 14121 \quad B = \frac{451872}{42} \approx 10759.33 \quad C = \frac{451872}{62} \approx 7288.26 \][/tex]
Therefore, the angles in triangle [tex]\(ABC\)[/tex] are approximately:
[tex]\[ A \approx 79^\circ.00, \quad B \approx 60^\circ.23, \quad C \approx 40^\circ.80 \][/tex]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.