1.3.2: Homework
- Page ID
- 197470
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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 two conditions must be met for two triangles to be congruent?
- What is the symbol used to indicate that two triangles are congruent?
- What is the key difference between congruent triangles and similar triangles?
- Are all congruent triangles similar? Are all similar triangles congruent?
- What does it mean for the corresponding sides of two triangles to be "proportional"?
- According to the Similarity Conditions theorem, what are the two ways to verify that two triangles are similar?
- If two right triangles share one pair of equal acute angles, are the triangles similar?
- In Example 2, how is the altitude of an equilateral triangle used to create two congruent right triangles?
- In Example 5, what assumption is made about the sun's light rays that allows us to conclude the triangles are similar?
- When triangles overlap, as in Example 6, what must you be careful about when determining the side lengths for your proportion?
Skills Refresher
The following is a set of review exercises you will need for this section.
For each of the following exercises, solve the equation.
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\(\dfrac{x}{12} = \dfrac{3}{x}\)
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\(1+\dfrac{x}{2} = \dfrac{2x}{5}\)
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\( \dfrac{x}{6} = \dfrac{1}{6} + \dfrac{x}{6} \)
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\( x + \dfrac{6}{x} = -5 \)
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\( \dfrac{5}{2x} = \dfrac{17}{18} - \dfrac{1}{3x} \)
- Answers
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\(x = \pm 6\)
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\(x = -10\)
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\( x = 4 \)
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\( x = -3 \) and \( x = -2 \)
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\( x = 3 \)
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Homework
Vocabulary Check
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If given two triangles that are the same size and shape, we say they are ___; however, if their size is different but their shape remains the same, we call them ___ triangles.
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The ___ of a triangle is the segment from one vertex of the triangle perpendicular to the ___ side.
Concept Check
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What is the difference between congruent triangles and similar triangles?
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What is the name of the short-cut method for solving proportions? Why does the method work?
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In two triangles, are the triangles similar if two corresponding pairs of angles are equal? How do you know?
True or False? For each of 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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The diagonal of a parallelogram splits the shape into two congruent triangles.
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In a \( 30^{ \circ } \)-\( 60^{ \circ } \)-\( 90^{ \circ } \) triangle, the leg opposite the \( 60^{ \circ } \) angle is half the length of the hypotenuse.
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If two right triangles have one pair of angles with the same measure, then the triangles are similar.
Basic Skills
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For the triangles shown, which of the following equations is true? Explain why.
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\(\dfrac{4}{x} = \dfrac{6}{8}\)
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\(\dfrac{x}{4} = \dfrac{6}{8}\)
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\(\dfrac{x}{x+4} = \dfrac{6}{8}\)
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\(\dfrac{x}{x+4} = \dfrac{6}{14}\)
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For each of the following exercises, decide whether the triangles are similar and explain why or why not.
For each of the following exercises, assume the triangles are similar. Solve for the variables. (Figures are not drawn to scale.)
For each of the following exercises, use properties of similar triangles to solve for the variable.
For each of the following exercises, use properties of similar triangles to solve for the variable.
Synthesis Questions
For each of the following exercises, name two congruent triangles and find the unknown quantities.
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\(P QRS\) is an isosceles trapezoid.
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\(\Delta PRU\) is isosceles.
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\(\Delta ARN\) is isosceles and \(OR = NG\). Find \(\angle RNG\) and \(\angle RNO\).
For each of the following exercises, explain why the measurements shown are inaccurate.
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- The follow pairs of triangles are similar. Solve for \(y\) in terms of \(x\).
For each of the following exercises, solve for \(y\) in terms of \(x\).
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Triangle \(ABC\) is a right triangle, and \(AD\) meets the hypotenuse \(BC\) at a right angle.
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If \(\angle ACB = 20^{\circ}\), find \(\angle B\), \(\angle CAD\), and \(\angle DAB\).
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Find two triangles similar to \(\Delta ABC\). List the corresponding sides in each of the triangles.
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Applications
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Measuring Distances. Ivan and Kim want to measure the distance across a stream. They mark point \(A\) directly across the stream from a tree at point \(T\) on the opposite bank. Ivan walks from point \(A\) down the bank a short distance to point \(B\) and sights the tree. He measures the angle between his line of sight and the stream bank.
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Draw a figure showing the stream, the tree, and the right triangle \(\triangle ABT\).
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Meanwhile, Kim, still standing at point \(A\), walks away from the stream at right angles to Ivan’s path. Ivan watches her progress and tells her to stop at point \(C\) when the angle between the stream bank and his line of sight to Kim is the same as the angle from the stream bank to the tree. Add triangle \(\triangle ABC\) to your figure.
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Ivan now measures the distance from point \(A\) to Kim at point \(C\). Explain why this distance is the same as the distance across the stream.
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Measuring Distances. If you have a baseball cap, here is another way to measure the distance across a river. Stand at point \(A\) directly across the river from a convenient landmark, say a large rock, on the other side. Tilt your head down so that the brim of the cap points directly at the base of the rock, \(R\).
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Draw a figure showing the river, the rock, and the right triangle \(\triangle ABR\), where \(B\) is the location of your baseball cap on your head.
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Now, without changing the angle of your head, rotate \(90^{\circ}\) and sight along the bank on your side of the river. Have a friend mark the spot \(C\) on the ground where the brim of your cap points. Add triangle \(\triangle ABC\) to your figure.
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Finally, you can measure the distance from point \(A\) to point \(C\). Explain why this distance is the same as the distance across the river.
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For each of the following exercises, use properties of similar triangles to solve.
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Heights. A rock climber estimates the height of a cliff she plans to scale. She places a mirror on the ground to see the top of the cliff in the mirror while she stands straight. The angles 1 and 2 formed by the light rays are equal, as shown in the figure. She then measures the distance to the mirror (2 feet) and the distance from the mirror to the base of the cliff (56 feet). How high is the cliff if she is 5 feet 6 inches tall?
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Heights. Lap wants to estimate the height of the Washington Monument. He notices that he can see the reflection of the top of the monument in the reflecting pool. He is 35 feet from the tip of the reflection, and that point is 1080 yards from the base of the monument, as shown below. From his physics class, Lap knows that the angles marked (in the image below) are equal. If Lap is 6 feet tall, what is his estimate for the height of the Washington Monument?
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Distances. In the sixth century BC, the Greek philosopher and mathematician Thales used similar triangles to measure the distance to a ship at sea. Two observers on the shore at points \(A\) and \(B\) would sight the ship and measure the angles formed, as shown in Figure (a). They would then construct a similar triangle, as shown in Figure (b), with the same angles at \(A\) and \(B\), and measure its sides. (This method is called triangulation.) Use the lengths given in the figures to find the distance from the observer at location \(B\) to the ship.
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Distances. The Capilano Suspension Bridge is a footbridge that spans a 230-foot gorge north of Vancouver, British Columbia. Before crossing the bridge, you decide to estimate its length. You walk 100 feet downstream from the bridge and sight its far end, noting the angle formed by your line of sight, as shown in Figure (a). You then construct a similar right triangle with a two-centimeter base, as shown in Figure (b). You find that the height of your triangle is 8.98 centimeters. How long is the Capilano Suspension Bridge?
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Surface Area. A conical tank is 12 feet deep, and the top's diameter is 8 feet. If the tank is filled with water to a depth of 7 feet, as shown in the figure at right, what is the area of the exposed surface of the water?
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Distances. To measure the distance \(EC\) across the lake shown in the figure at right, stand at \(A\) and sight point \(C\) across the lake, then mark point \(B\). Then sight to point \(E\) and mark point \(D\) so that \(DB\) is parallel to \(EC\). If \(AD = 25\) yards, \(AE = 60\) yards, and \(BD = 30\) yards, how wide is the lake?
Challenge Problems
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Distances. Here is a way to find the distance across a gorge using a carpenter’s square and a five-foot pole. Plant the pole vertically on one side of the gorge at point \(A\) and place the angle of the carpenter’s square on top of the pole at point \(B\), as shown in the figure. Sight along one side of the square so that it points to the opposite side of the gorge at point \(P\). Without moving the square, sight along the other side and mark point \(Q\). If the distance from \(Q\) to \(A\) is six inches, calculate the width of the gorge. Explain your method.
