Which Law or Theorem Would You Use? Pythagorean Theorem Why? Triangle △ ABC is a right triangle. Remaining Side and Angles: b≈ 16.2, m∠ A≈ 67.8^(∘), m∠ C≈ 22.2^(∘)
Practice makes perfect
Let's start by sketching △ ABC and labeling sides a, c, and angle ∠ B. Remember that m∠ B= 90^(∘), a= 15, and c= 6.
We will solve △ ABC. This means we will find the values of b, m∠ A, and m∠ C. First, let's find the length of the remaining side and then let's move on to the measures of the remaining angles.
Remaining Side
Since m∠ B=90^(∘), △ ABC is a right triangle. We are given the lengths of two sides of the triangle, so we can use the Pythagorean Theorem to find the length b of the third side. Let's recall the theorem.
Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.
In our case the length of the hypotenuse is b, and the lengths of the legs are a= 15 and c= 6. Let's apply the Pythagorean Theorem to our triangle.
15^2+ 6^2=b^2
Finally we will solve the obtained equation for b.
The length b of the third side of the triangle is about 16.2 units. Let's update our diagram.
Remaining Angles
Next we will find the measures of the two remaining angles. Let's start with m∠ A. Since now we know the lengths of all three sides of △ ABC and the measure of one angle, we can use the Law of Sines.
Law of Sines
For any triangle the ratio between the sine value of an angle and the length of its opposite side is constant.
Let's apply the Law of Sines to △ ABC.
sinA/15=sin 90^(∘)/16.2=sinC/6
Now we will write an equation involving the sine of ∠ A.
sinA/15=sin90^(∘)/16.2 ⇔ sinA=15sin90^(∘)/16.2
To find the measure of ∠ A, let's use the inverse sine ratio.
sinA=15sin90^(∘)/16.2
⇕
m∠ A=sin^(- 1)(15sin90^(∘)/16.2)
We will use a calculator to approximate m∠ A to the nearest tenth of a degree.
m∠ A=sin^(- 1)(15sin90^(∘)/16.2)≈ 67.8^(∘)
To find the measure of ∠ C, we could solve an equation involving the sine of ∠ C, or we can use the Triangle Sum Theorem.
m∠ C≈ 180^(∘)-67.8^(∘)- 90^(∘)=22.2^(∘)
The value of m∠ C is about 22.2^(∘).
