Start with recalling that the sum of the angle measures in a triangle is 180^(∘).
Angle Measures
Side Lengths
m∠ A
53^(∘)
AB
12
m∠ B
90^(∘)
BC
16
m∠ C
37^(∘)
AC
20
Practice makes perfect
We are given a triangle and asked to solve it, which means we need to find all angle measures and all side lengths.
Let's recall that the sum of the angle measures in a triangle is 180^(∘). Using this information, we can create an equation.
(4x+1)^(∘)+ (7x-1)^(∘)+ (3x-2)^(∘)=180^(∘)
Next we will solve the equation to find the value of x.
The value of x is 13. Now we will find the angle measures of △ ABC using this value.
Angle Measure
Substitute
Simplify
m∠ A
4(13)+1
53^(∘)
m∠ B
7(13)-1
90^(∘)
m∠ C
3(13)-2
37^(∘)
Let's add the angle measures to our picture. Notice that we found that ∠ B is a right angle.
Since △ ABC is a right triangle, we can use the trigonometric ratios to find the side lengths. Let's recall the definition of the sine of an angle.
If△ ABCis a right triangle with acute∠ A,
then the sine of∠ Ais the ratio of the length
of the leg opposite∠ Ato the length of
the hypotenuse.
Using this theorem, we can find an equation for sin 53^(∘). The length of the leg opposite to this angle is 2y+2 and the hypotenuse has a length of 3y-1.
sin 53^(∘)=2y+2/3y-1
Let's solve above equation using the fact that the approximate value of sin 53^(∘) is 0.8.
