1.2.2: Homework
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Skills Refresher
The following is a set of review exercises you will need for this section.
For the following exercises, solve the inequality.
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\(6-x>3\)
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\(\dfrac{-3 x}{4} \geq-6\)
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\(3 x-7 \leq-10\)
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\(4-3 x<2 x+9\)
For the following exercises, assume that \(x<0\). If this is the case, is the expression positive or negative?
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\(-x\)
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\(-(-x)\)
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\(|x|\)
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\(-|x|\)
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\(-|-x|\)
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\(x^{-1}\)
- Answers
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\(x<3\)
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\(x \leq 8\)
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\(x \leq -1\)
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\(x > -1\)
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Positive
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Negative
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Positive
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Negative
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Negative
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Negative
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Homework
Reading Questions
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What is the sum of the interior angles in any triangle?
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What does the Triangle Inequality Theorem state about the side lengths of a triangle?
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What is the definition of an equilateral triangle?
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In an isosceles triangle, which angles are referred to as the "base angles," and what is true about their measures?
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What is the name for the side opposite the right angle in a right triangle?
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State the Pythagorean Theorem using the terms hypotenuse and legs.
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If you are given the three side lengths of a triangle, how can you test if it is a right triangle?
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Is it possible for a triangle to contain two right angles? Why or why not?
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In a 30°-60°-90° triangle, which side is the longest?
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What is the relationship between the hypotenuse and the shortest side in a 30°-60°-90° triangle?
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What is another name for a 45°-45°-90° triangle?
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If the two legs of a 45°-45°-90° triangle each have length \(a\), what is the length of the hypotenuse?
Vocabulary Check
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A three-sided polygon is another way of saying ___.
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(MyOpenMath) If given the triangle \( \triangle ABC \), it is common notation to label the side opposite \( \angle A \) with the letter ___.
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(MyOpenMath) A triangle where all three sides have the same length is called a ___ triangle.
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An isosceles triangle has two ___ angles.
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The ___ angle of an isosceles triangle lies between the two sides of equal length.
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(MyOpenMath) A(n) ___ triangle must have one angle ___ \( 90^{ \circ } \).
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A(n) ___ triangle must have all angles ___ \( 90^{ \circ } \).
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The two special triangles are the ___-\( 90^{ \circ } \) and ___-\( 90^{ \circ } \) triangles.
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Another name for the \( 45^{ \circ } \)-\( 45^{ \circ } \)-\( 90^{ \circ } \) triangle is a(n) ___ triangle.
Concept Check
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Is it always true that the hypotenuse is the longest side in a right triangle? Why or why not?
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In \(\triangle D E F\), is it possible that \(d+e>f\) and \(e+f>d\) are both true? Explain your answer.
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In a right triangle with hypotenuse \(c\), we know that \(a^2+b^2=c^2\). Is it also true that \(a+b=c\)? Why or why not?
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Is having more than one obtuse angle in a triangle possible? Why or why not?
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Draw any quadrilateral (a four-sided polygon) and divide it into two triangles by connecting two opposite vertices by a diagonal. What is the sum of the angles in your quadrilateral?
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What is the difference between a vertex and vertical angles?
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In any triangle, the sum of the three angles is ___.
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The longest side in a triangle is opposite the ___ angle, and the shortest side is opposite the ___ angle.
True or False? For the following exercises, determine if the statement is true or false. If true, cite the definition or theorem stated in the text supporting your claim. If false, explain why it is false and, if possible, correct the statement.
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A right triangle always has one angle of \(90^{\circ}\).
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Given a triangle with side lengths \( a \), \( b \), and \( c \), where \( c \) is the longest side,\[a^2 + b^2 = c^2.\nonumber \]
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All of the angles of an equilateral triangle are equal.
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The vertex angles of an isosceles triangle are equal.
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If the sum of the lengths of any two sides is greater than the length of the third side, then the triangle must be a right triangle.
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In \( \triangle ABC \), \(a^2 + b^2 = c^2\).
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If the sides of a triangle satisfy the relationship \(a^2 + b^2 = c^2\), then the triangle is a right triangle.
Basic Skills
For the following exercises, sketch and label a triangle with the given properties.
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An isosceles triangle with vertex angle \(30^{\circ}\)
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A scalene triangle with one obtuse angle
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(MyOpenMath) A right triangle with legs 4 and 7
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(MyOpenMath) A right triangle with hypotenuse 12, where one leg is of length 8.
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An isosceles right triangle
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An isosceles triangle with one obtuse angle
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A right triangle with one angle \(20^{\circ}\)
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(MyOpenMath) The triangle \( \triangle ABC \) has \( m \angle A = 88^{\circ} \) and \( m \angle B = 87^{ \circ } \), find \( m \angle C \).
For the following exercises, find the unknown angle.
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If two sides of a triangle are 6 feet and 10 feet long, what are the largest and smallest possible values for the length of the third side?
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Two adjacent sides of a parallelogram are 3 cm and 4 cm long. What are the largest and smallest possible values for the length of the diagonal?
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If one of the equal sides of an isosceles triangle is 8 millimeters long, what are the largest and smallest possible values for the length of the base?
For the following exercises, find the unknown side of the triangle.
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Find \(\alpha\) and \(\beta\).
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(MyOpenMath) Find \(x\) and \(y\).
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(MyOpenMath) Find \(x\) and \(y\).
For the following exercises, decide whether a triangle with the given sides is a right triangle.
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(MyOpenMath) 9 in, 16 in, 25 in
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12 m, 16 m, 20 m
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5 m, 12 m, 13 m
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5 ft, 8 ft, 13 ft
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\(5^2\) ft, \(8^2\) ft, \(13^2\) ft
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\(\sqrt{5}\) ft, \(\sqrt{8}\) ft, \(\sqrt{13}\) ft
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(MyOpenMath) Compute the exact value of the altitude \( h \) in an equilateral triangle, where all sides are 14 inches long.
Synthesis Questions
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(MyOpenMath) In a certain triangle, angle \( A \) is three times as big as angle \( C \) and angle \( B \) is \( 55^{ \circ } \) more than angle \( C \). What is the degree measure of angle \( B \)?
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The two shorter sides of an obtuse triangle are 3 in and 4 in. What are the possible lengths for the third side?
For the following exercises, explain why the measurements shown cannot be accurate.
For the following exercises, find the unknown angle. Arrows on a pair of lines indicate that they are parallel.
For the following exercises, arrows on a pair of lines indicate that they are parallel. Find \(x\) and \(y\).
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How long is the diagonal of a square whose side is 6 centimeters?
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How long is the side of a cube whose volume is 80 cubic feet?
For the following exercises, find the value of \( x \).
For the following exercises, find the length of the indicated side(s).
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(MyOpenMath) \( x \) and \( y \)
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(MyOpenMath) \( \overline{TY} \) and \( \overline{TX} \)
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(MyOpenMath) \( \overline{AY} \) and \( \overline{YB} \)
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(MyOpenMath) \( x \) and \( y \)
For the following exercises, the angle labeled \(\phi\) is called an exterior angle of the triangle, formed by one side and the extension of an adjacent side. Find \(\phi\).
For the following exercises, make a sketch of the situation described and label a right triangle. Then use the Pythagorean Theorem to solve the problem.
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The diagonal of a square is 12 inches long. How long is the side of the square?
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The length of a rectangle is twice its width, and its diagonal is meters long. Find the dimensions of the rectangle.
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(MyOpenMath) The hypotenuse of right triangle is 146 centimeters long. The difference between the other two sides is 14 centimeters. Find the exact values of the missing sides.
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(MyOpenMath) The shortest side of a right triangle is 9 centimeters long. The difference between the lengths of the other two sides is 1 centimeter. Find the exact values of the missing side and hypotenuse.
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(MyOpenMath) The shaded, half circular disk is drawn on the side of a right triangle, whose legs are labeled 5.3 meters and 10.1 meters. Find the area of this half disk. Round your answer to one decimal place.
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(MyOpenMath) Imagine that for the right triangle shown below, each of its three sides is bent into a perfect circle. What would be the combined total area of all three circles? Round your answer to one decimal place.
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What size rectangle can be inscribed in a circle of radius 30 feet if the length of the rectangle must be three times its width?
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What size square can be inscribed inside a circle of radius 8 inches, so that its vertices just touch the circle?
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Find \(\alpha, \beta\) and \(h\).
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Find \(\alpha, \beta\) and \(d\).
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(MyOpenMath) Find the slant height (labeled \( x \)) of the right circular cone.
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(MyOpenMath) Compute the radius of the base of the right circular cone if given the slant height of 20 cm and the cone height of 16 cm.
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(MyOpenMath) A right square pyramid is shown. The height of the pyramid is 8 units. The distance from the center of the base of the pyramid to vertex B is 15 units, as shown. Find the length of segment \( \overline{AB} \).
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(MyOpenMath) Compute the exact value of the height \( h \) of the square-based straight pyramid, given that the base is a square with sides \( a = 36 \) inches long, and all other edges are \( b = 54 \) inches long.
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Find the diagonal of a cube of side 8 inches. Hint: Find the diagonal of the base first.
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Find the diagonal of a rectangular box whose sides are 6 cm by 8 cm by 10 cm. Hint: Find the diagonal of the base first.
For the following exercises, \(\Delta ABC\) is equilateral. Find the unknown angles.
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Consider the following figure.
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Compute \(2\theta + 2\phi\).
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Compute \(\theta + \phi\).
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What must be true about \(\triangle ABC\)?
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Consider the following figure.
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Explain why \(\angle O A B\) and \(\angle A B O\) are equal in measure.
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Explain why \(\angle O B C\) and \(\angle B C O\) are equal in measure.
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Explain why \(\angle A B C\) is a right angle.
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In the figure below, find \(\theta\), and justify your answer.
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Write an algebraic expression for \(\theta\) in the figure below.
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Applications
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Bounding Distances. The town of Madison is 15 miles from Newton, and 20 miles from Lewis. What are the possible values for the distance from Lewis to Newton?
For the following exercises, make a sketch of the situation described and label a right triangle. Then use the Pythagorean Theorem to solve the problem.
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TV Dimensions. The size of a TV screen is the length of its diagonal. If the width of a 35-inch TV screen is 28 inches, what is its height?
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TV Dimensions (MyOpenMath). Piyali goes to the store to buy a new TV. She tells the salesperson that she needs a TV that has a maximum width of 37 inches and a maximum height of 28 inches. Assuming that she can get a TV with these maximum dimensions, and knowing that TVs are measured by the diagonal of the screen, what size TV can Piyali buy?
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Tree Shadows. If a 30-meter pine tree casts a shadow of 30 meters, how far is the tip of the shadow from the top of the tree?
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Utility Pole (MyOpenMath). A utility pole is 9 m high. A cable is stretched from the top of the pole to a point in the ground that is 8 m from the bottom of the pole. How long is the cable?
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Distance Between Towns (MyOpenMath). If Nhat is 22 miles due west of Sacramento and Loi is 30 miles due north of Sacramento, what is the shortest distance from Loi to Nhat?
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Altitude (MyOpenMath). What is the altitude of the plane?
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Ladder Length (MyOpenMath). What is the minimum length the ladder must be to reach the roof?
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A 24-foot flagpole is being raised by a rope and pulley, as shown in the figure. The loose end of the rope can be secured to a ring on the ground 7 feet from the base of the pole. From the ring to the top of the pulley, how long should the rope be when the flagpole is vertical?
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To check whether the corners of a frame are square, carpenters sometimes measure the sides of a triangle, with two sides meeting at the join of the boards. Is the corner shown in the figure square?
For the following exercises, make a sketch and solve.
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Pesky Pipe. The back of Ron’s pickup truck is five feet wide and seven feet long. He wants to bring home a 9-foot length of copper pipe. Will it lie flat on the floor of the truck?
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Pesky Pipe. Mary has a pickup truck with a camper shell. The camper shell floor is five feet wide and seven feet long, but the shell is only three feet tall. Will a 9-foot copper pipe fit diagonally across the back of the truck?
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What is the longest curtain rod that will fit inside a box 60 inches long by 10 inches wide by 4 inches tall?
Challenge Problems
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(MyOpenMath) A sidewalk goes all the way around a rectangular field, as shown below (not drawn to scale). The field is 65 feet long and 45 feet wide. You need to get from the bottom-left corner to the top-right corner. Your walking speed is 7 feet per second along the sidewalk and 4 feet per second through the field.
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How many seconds will it take you if you walk from the bottom-left corner to the top-right corner directly across the field?
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How many seconds will it take you if you walk all 65 feet along the sidewalk (from bottom-left to bottom-right) and then 45 feet along the sidewalk (from bottom-right to top-right)?
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Suppose you walk \( x \) feet along the sidewalk and then turn to walk the remaining distance through the field directly to the top-right corner. Write an expression in \( x \) representing the number of seconds it would take you to walk this path.
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(MyOpenMath) Find the values of \( x \), \( y \), and \( z \) to two decimal places.
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(MyOpenMath) Find the values of \( x \), \( y \), and \( z \) to two decimal places.
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(MyOpenMath) Find the values of \( x \), \( y \), and \( z \) to two decimal places.
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(MyOpenMath) Find the values of \( x \), \( y \), and \( z \) to two decimal places.
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(MyOpenMath) Find the values of \( x \), \( y \), and \( z \) to two decimal places.
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(MyOpenMath) An arch is in the shape of a semicircle. At a point along the base 4 feet from an end of the arch, the height of the arch is 8 feet. Find the exact value of the maximum height of the arch.
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Consider the following figure.
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Compare \(\theta\) with \(\alpha+\beta\). (Hint: What do you know about supplementary angles and the sum of angles in a triangle?
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Compare \(\alpha\) and \(\beta\).
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Explain why the inscribed angle \(\angle B A O\) is half the size of the central angle \(\angle B O C\).
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Find an algebraic expression for \(\phi\) and use your answer to write a rule for finding an exterior angle of a triangle.
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Find the three exterior angles of the triangle. What is the sum of the exterior angles?
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Write an algebraic expression for each exterior angle in terms of one of the angles of the triangle. What is the sum of the exterior angles?
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(MyOpenMath) The area of square \( ABCD \) is one square unit. Triangle \( \triangle PCQ \) is equilateral. What are the exact lengths of the sides of the triangle?
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Proving the Pythagorean Theorem. There are many proofs of the Pythagorean Theorem. Here is a simple visual argument.
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What is the length of the side of the large square in the figure? Write an expression for its area.
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Write another expression for the area of the large square by adding the areas of the four right triangles and the smaller central square.
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Equate your two expressions for the area of the large square, and deduce the Pythagorean Theorem.
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For the following exercises, the figures inscribed in each problem are regular polygons, which means that all their sides are the same length, and all the angles have the same measure. Find the angles \(\theta\) and \(\phi\).
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A tangent meets the radius of a circle at a right angle. In the figure, \(\angle AOB = 140^{\circ}\). Find the angles labeled 1 through 5 in order, and give a reason for each answer.
