Exercise 4 Page 865

Let's analyze the given right triangle.

We will find the missing measures one at a time. In this case, this means that we want to find m ∠ A, m∠ B, and b.

Side Length

a^2 + b^2 = c^2 Let's substitute the given values and solve for b.

a^2 + b^2 = c^2

SubstituteValues

Substitute values

9^2 + b^2 = 12^2

▼

Solve for b

CalcPow

Calculate power

81 + b^2 = 144

SubEqn

LHS-81=RHS-81

b^2 = 63

SqrtEqn

sqrt(LHS)=sqrt(RHS)

b = sqrt(63)

UseCalc

Use a calculator

b = 7.937254...

RoundDec

Round to 1 decimal place(s)

b ≈ 7.9

Since the length of the side of the triangle cannot be negative, we only consider the positive solution.

Angle Measures

We can find m ∠ A using a sine ratio. Sine A=Opposite ∠ A/Hypotenuse ⇒ sin A = a/c By the definition of the inverse sine, the inverse sine of 912 is the measure of ∠ A. To find it, we have to use a calculator.

m ∠ A = sin^(- 1) 9/12

UseCalc

Use a calculator

m ∠ A = 48.590378...

RoundInt

Round to nearest integer

m ∠ A ≈ 49^(∘)

What is more, according to the Rules:Interior Angles Theorem the sum of all angles in any given triangle is 180^(∘). Therefore we can find m ∠ B. m ∠ A + m∠ B + m ∠ C = 180^(∘) Let's substitute the given values and solve for m ∠ B.

m ∠ A + m ∠ B + m ∠ C = 180^(∘)

SubstituteII

m ∠ A= 49^(∘), m∠ C= 90^(∘)

49 ^(∘) + m ∠ B + 90^(∘) = 180^(∘)

▼

Solve for m∠ B

AddTerms

Add terms

139^(∘) + m ∠ B = 180^(∘)

SubEqn

LHS-139^(∘)=RHS-139^(∘)

m ∠ B = 41^(∘)

Because m ∠ A was approximately 49^(∘), therefore the measure of angle B is also an approximation, m ∠ B ≈ 41^(∘).

Solved Triangle

Finally, let's gather all of our findings.