Answer :
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]
