Chapter Closure
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Exercise 116 Page 477
a Since we know two angles, we can determine the third angle using the Triangle Angle Sum Theorem.
b What trigonometric ratio can be used to determine x?
c What theorem can you use to calculate an unknown side in a right triangle?
d What trigonometric ratio can be used to determine x?
e What theorem can you use to calculate an unknown side in a right triangle?
a x=8 cm
b x≈ 30.87 feet
c x≈ 15.20 inches
d x≈3.75 units
e x = 7sqrt(2) units
a Two of the triangle's angles are 60^(∘). By the Triangle Angle Sum Theorem, we can determine the third angle.
θ + 60^(∘)+60^(∘)=180^(∘) ⇔ θ=60^(∘) Since all of the triangle's angles are 60^(∘), this is an equilateral triangle. In an equilateral triangle, all three sides have the same length. Therefore, we know that x=8 cm.
b From the diagram, we have been given the adjacent leg to the reference angle and are asked to find the length of the hypotenuse. With this information, we can use the cosine ratio to determine x.
Let's look at the given diagram and substitute the given values into the above equation.
c Let's take a look at the given diagram.
It depicts a right triangle in which the hypotenuse and one of the legs is given. We want to find the length of the other leg, which we can do using the Pythagorean Theorem.
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Pythagorean Theorem |
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In a right triangle, the length of the hypotenuse squared equals the sum of the squares of the lengths of the legs. a^2+b^2=c^2 |
d From the diagram, we have been given the opposite leg to the reference angle and are asked to find the length of the adjacent one. With this information, we can use the tangent ratio to determine x.
Let's look at the given diagram and substitute the given values into the above equation.
e Let's take a look at the given diagram.
It depicts a right triangle in which the the two legs have the same length of 7 units. We want to find the length of the hypotenuse, which we can do using the Pythagorean Theorem.
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Pythagorean Theorem |
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In a right triangle, the length of the hypotenuse squared equals the sum of the squares of the lengths of the legs. a^2+b^2=c^2 |
a^2+b^2=c^2
SubstituteValues
Substitute values
7^2+ 7^2= x^2
▼
Solve for x
CalcPow
Calculate power
49+49=x^2
SumToProdTwoTerms
a+a=2a
2(49)=x^2
SqrtEqn
sqrt(LHS)=sqrt(RHS)
sqrt(2(49))=x
SqrtProd
sqrt(a* b)=sqrt(a)*sqrt(b)
sqrt(2)sqrt(49)=x
CalcRoot
Calculate root
sqrt(2)(7)=x
CommutativePropMult
Commutative Property of Multiplication
7sqrt(2)=x
RearrangeEqn
Rearrange equation
x = 7sqrt(2)
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