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]
