11.3.2: Homework
- Page ID
- 197630
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Reading Questions
- What three pieces of information about a triangle are required to use the area formula \(A = \frac{1}{2}ab\sin(\theta)\)?
- In the formula \(A = \frac{1}{2}ab\sin(\theta)\), what is the specific relationship between the angle \(\theta\) and the sides \(a\) and \(b\)?
- What information about a triangle is required to use Heron's Formula?
- What does the variable \(s\) represent in Heron's Formula, and how is it calculated?
- How can you find the area of an irregular quadrilateral using the methods from this section?
- The proof of Heron's Formula provided in the text relies on which two other major triangle formulas?
Skills Refresher
Review the following skills you will need for this section.
For the following exercises, find the area of the triangle.
- Answers
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- 24
- 24
- 24
- 24
Homework
Concept Check
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Explain what \( s \) represents in Heron’s formula.
Basic Skills
For the following exercises, find an exact value for the area of the triangle.
For the following exercises, find the area of the triangle. Round your answer to the nearest tenth.
For the following exercises,
(a) Find exact values for the base and height of the triangle.
(b) Compute an exact value for the area of the triangle.
For the following exercises, find the area of the triangle with the given properties. Round to the nearest tenth.
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(MyOpenMath) one side has length 37 feet, another side has length 11 feet, and the included angle measures \( 51^{ \circ } \).
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\(b=2.5\) in, \(c=7.6\) in, \(A=138^{\circ}\)
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\(a=0.8 \mathrm{~m}, c=0.15 \mathrm{~m}, B=15^{\circ}\)
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the sides of the triangle are 27 cm, 15 cm, and 20 cm.
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the side lengths are 18 in, 21 in, and 32 in.
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the triangle has sides of length 20 cm, 26 cm, and 37 cm.
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(MyOpenMath) An isosceles triangle has an area of \( 168 \) square feet, and the angle between the two sides of equal length is \( 37^{ \circ } \). Find the length of one of the two equal sides. Round your answer to one decimal place.
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(MyOpenMath) A triangle has an area of \( 352 \) square meters, and two sides of the triangle have lengths 42 meters and 19 meters. Find the angle included by these two sides. Find the length of one of the two equal sides. Round your answer to one decimal place.
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Write an expression for the area of the triangle.
Synthesis Questions
- A parallelogram has sides of length 15.4 units and 9.8 units. Its area is 72.9 square units. Find the measure of the longer diagonal.
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The four sequential sides of a quadrilateral have lengths 4.5 cm, 7.9 cm, 9.4 cm, and 12.9 cm. The angle between the two smallest sides is \( 117^{ \circ } \). What is the area of this quadrilateral?
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The four sequential sides of a quadrilateral have lengths 5.7 cm, 7.2 cm, 9.4 cm, and 12.8 cm. The angle between the two smallest sides is \( 106^{ \circ } \). What is the area of this quadrilateral?
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Find the area of the regular pentagon shown below. (Hint: The pentagon can be divided into five congruent triangles.)
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Find the perimeter of a regular hexagon if the apothegm is 8 cm long. (The apothegm is the segment from the center of the hexagon and perpendicular to one of its sides.)
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Find the area of the hexagon.
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Find the area of the regular hexagon shown below. (Hint: The hexagon can be divided into six congruent triangles.)
Applications
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Find the area of a triangular piece of land that measures 30 feet on one side and 42 feet on another; the included angle measures 132°. Round to the nearest whole square foot.
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Find the area of a triangular piece of land that measures 110 feet on one side and 250 feet on another; the included angle measures 85°. Round to the nearest whole square foot.
For the following exercises, lots from a housing development have been subdivided into triangles. Find the total area of each lot by computing and adding the areas of each triangle.
