To solve the right triangle ABC where angle C is the right angle (90°), we are given the following:
- Angle B = 70.0°
- Side b = 112 units
- Angle A = 20.0°
We aim to find the lengths of sides [tex]\( a \)[/tex] (opposite to angle A) and [tex]\( c \)[/tex] (the hypotenuse).
Given:
Angle A + Angle B + Angle C = 180°
Angle A = 20°
Angle B = 70°
Angle C = 90°
To find side [tex]\( a \)[/tex]:
1. Trigonometric Relation Using Tangent: In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side:
[tex]\[ \tan(\text{Angle A}) = \frac{\text{Opposite side (a)}}{\text{Adjacent side (b)}} \][/tex]
2. Solving for side [tex]\( a \)[/tex]:
[tex]\[ a = b \cdot \tan(\text{Angle A}) \][/tex]
Substituting the given values:
[tex]\[ a = 112 \cdot \tan(20^\circ) \][/tex]
From the calculations, the length of side [tex]\( a \)[/tex] is approximately:
[tex]\[ a \approx 40.8 \text{ units} \][/tex]
(Rounded to the nearest tenth)
Next, to find side [tex]\( c \)[/tex] (hypotenuse):
3. Using the Pythagorean Theorem: In a right triangle:
[tex]\[ c^2 = a^2 + b^2 \][/tex]
4. Substituting the known values:
[tex]\[ c = \sqrt{a^2 + b^2} \][/tex]
[tex]\[ c = \sqrt{(40.8)^2 + (112.0)^2} \][/tex]
[tex]\[ c \approx 119.2 \text{ units} \][/tex]
(Rounded to the nearest tenth)
To summarize:
- The length of side [tex]\( a \)[/tex] is approximately [tex]\( 40.8 \)[/tex] units.
- The length of side [tex]\( c \)[/tex] is approximately [tex]\( 119.2 \)[/tex] units.
