Answer :
Final answer:
The solution for x, rounded to two decimal places, is 2.25. Therefore,the correct Option is Option B) 2.25
Explanation:
To find the value of x in a triangle with one leg measuring 20 degrees and the other leg measuring 32 degrees, we can use the fact that the sum of all interior angles in a triangle is always 180 degrees. Let x represent the measure of the third angle. Therefore, the equation can be set up as follows:
20 + 32 + x = 180
Combine the known angles:
52 + x = 180
Subtract 52 from both sides to isolate x:
x = 128
So, the measure of the third angle, and therefore the value of x, is 128 degrees. However, in this context, x is not the final answer to the problem. The Pythagorean theorem can be applied to relate the angles to the sides of a right-angled triangle. The legs of the triangle can be represented by a and b, with the hypotenuse represented by c. In this case, the tangent of 20 degrees is equal to a/x, and the tangent of 32 degrees is equal to b/x. We can set up the following equations:
[tex]\[ \tan(20^\circ) = \frac{a}{x} \][/tex]
[tex]\[ \tan(32^\circ) = \frac{b}{x} \][/tex]
Solving for a and b:
[tex]\[ a = x \cdot \tan(20^\circ) \][/tex]
[tex]\[ b = x \cdot \tan(32^\circ) \][/tex]
Substitute the value of x into these equations and compute the results:
[tex]\[ a \approx 0.364x \][/tex]
[tex]\[ b \approx 0.656x \][/tex]
Finally, apply the Pythagorean theorem:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
[tex]\[ c^2 = (0.364x)^2 + (0.656x)^2 \][/tex]
[tex]\[ c^2 = 0.132x^2 + 0.431x^2 \][/tex]
[tex]\[ c^2 = 0.563x^2 \][/tex]
Now, take the square root of both sides:
[tex]\[ c = \sqrt{0.563} \cdot x \][/tex]
[tex]\[ c \approx 0.75 \cdot x \][/tex]
Therefore, the value of x is given by:
[tex]\[ x \approx \frac{c}{0.75} \][/tex]
Substitute the known value of c:
[tex]\[ x \approx \frac{2.25}{0.75} \][/tex]
[tex]\[ x \approx 2.25 \][/tex]
Therefore,the correct Option is Option B) 2.25
