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# 5.3: Identify Accurate Drawings of Triangles

Difficulty Level: At Grade Created by: CK-12
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Practice Identify Accurate Drawings of Triangles

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Your friend is creating a new board game that involves several different triangle shaped pieces. However, the game requires accurate measurements of several different pieces that all have to fit together. She brings some of the pieces to you and asks if you can verify that her measurements of the pieces' side lengths and angles are correct.

You take out the first piece. According to your friend, the piece has sides of length 4 in, 5 in and 7 in, and the angle between the side of the length 4 and the side of length 5 is 78\begin{align*}78^\circ\end{align*}. She's very confident in the lengths of the sides, but not quite sure if she measured the angle correctly. Is there a way to determine if your friend's game piece has the correct measurements, or did she make a mistake?

It is indeed possible to determine if your friend's measurements are correct or not. At the end of this Concept, you'll be able to tell your friend if her measurements were accurate.

### Guidance

Our extension of the analysis of triangles draws us naturally to oblique triangles. The Law of Cosines can be used to verify that drawings of oblique triangles are accurate. In a right triangle, we might use the Pythagorean Theorem to verify that all three sides are the correct length, or we might use trigonometric ratios to verify an angle measurement. However, when dealing with an obtuse or acute triangle, we must rely on the Law of Cosines.

#### Example A

In ABC\begin{align*}\triangle{ABC}\end{align*} at the right, a=32,b=20\begin{align*}a = 32, b = 20\end{align*}, And c=16\begin{align*}c = 16\end{align*}. Is the drawing accurate if it labels C\begin{align*}\angle{C}\end{align*} as 35.2\begin{align*}35.2^\circ\end{align*}? If not, what should C\begin{align*}\angle{C}\end{align*} measure?

Solution: We will use the Law of Cosines to check whether or not C\begin{align*}\angle{C}\end{align*} is 35.2\begin{align*}35.2^\circ\end{align*}.

162256256256=202+3222(20)(32)cos35.2=400+10242(20)(32)cos35.2=400+10241045.94547378.05453Law of CosinesSimply squaresMultiplyAdd and subtract\begin{align*}16^2 & = 20^2 + 32^2 - 2(20)(32) \cos 35.2 && \text{Law of Cosines} \\ 256 & = 400 + 1024 - 2(20)(32) \cos 35.2 && \text{Simply squares} \\ 256 & = 400 + 1024 - 1045.94547 && \text{Multiply} \\ 256 & \neq 378.05453 && \text{Add and subtract}\end{align*}

Since 256378.05453\begin{align*}256 \neq 378.05453\end{align*}, we know that C\begin{align*}\angle{C}\end{align*} is not 35.2\begin{align*}35.2^\circ\end{align*}. Using the Law of Cosines, we can figure out the correct measurement of C\begin{align*}\angle{C}\end{align*}.

16225625625611680.912524.1=202+3222(20)(32)cosC=400+10242(20)(32)cosC=400+10241280cosC=14241280cosC=1280cosC=cosCCLaw of CosinesSimplify SquaresMultiplyAddSubtract 1424Dividecos1(0.9125)\begin{align*}16^2 & = 20^2 + 32^2 -2(20)(32) \cos C && \text{Law of Cosines} \\ 256 & = 400 + 1024 - 2(20)(32) \cos C && \text{Simplify Squares} \\ 256 & = 400 + 1024 - 1280 \cos C && \text{Multiply} \\ 256 & = 1424 - 1280 \cos C && \text{Add} \\ -1168 & = -1280 \cos C && \text{Subtract}\ 1424 \\ 0.9125 & = \cos C && \text{Divide} \\ 24.1^\circ & \approx \angle {C} && \cos^{-1}(0.9125) \end{align*}

For some situations, it will be necessary to utilize not only the Law of Cosines, but also the Pythagorean Theorem and trigonometric ratios to verify that a triangle or quadrilateral has been drawn accurately.

#### Example B

A builder received plans for the construction of a second-story addition on a house. The diagram shows how the architect wants the roof framed, while the length of the house is 20 ft. The builder decides to add a perpendicular support beam from the peak of the roof to the base. He estimates that new beam should be 8.3 feet high, but he wants to double-check before he begins construction. Is the builder’s estimate of 8.3 feet for the new beam correct? If not, how far off is he?

Solution: If we knew either A\begin{align*}\angle{A}\end{align*} or C\begin{align*}\angle{C}\end{align*}, we could use trigonometric ratios to find the height of the support beam. However, neither of these angle measures are given to us. Since we know all three sides of ABC\begin{align*}\triangle{ABC}\end{align*}, we can use the Law of Cosines to find one of these angles. We will find A\begin{align*}\angle{A}\end{align*}.

1421961963480.72543.5=122+2022(12)(20)cosA=144+400480cosA=544480cosA=480cosA=cosAALaw of CosinesSimplifyAddSubtractDividecos1(0.725)\begin{align*}14^2 & = 12^2 + 20^2 - 2(12)(20) \cos A && \text{Law of Cosines} \\ 196 & = 144 + 400 - 480 \cos A && \text{Simplify} \\ 196 & = 544 - 480 \cos A && \text{Add} \\ -348 & = -480 \cos A && \text{Subtract} \\ 0.725 & = \cos A && \text{Divide} \\ 43.5^\circ & \approx \angle{A} && \cos^{-1} (0.725) \end{align*}

Now that we know A\begin{align*}\angle{A}\end{align*}, we can use it to find the length of BD\begin{align*}BD\end{align*}.

sin43.512sin43.58.3=x12=xx\begin{align*}\sin 43.5 & = \frac{x}{12} \\ 12 \sin 43.5 & = x \\ 8.3 & \approx x\end{align*}

Yes, the builder’s estimate of 8.3 feet for the support beam is accurate.

#### Example C

In CIR,c=63,i=52\begin{align*}\triangle{CIR}, c = 63, i = 52\end{align*}, and r=41.9\begin{align*}r = 41.9\end{align*}. Find the measure of all three angles.

Solution:

632=522+41.9225241.9cosC522=632+41.9226341.9cosI18083.555.1=41.4C83.5I55.1R41.4\begin{align*} 63^2 = 52^2 + 41.9^2 - 2 \cdot 52 \cdot 41.9 \cdot \cos C \\ 52^2 = 63^2 + 41.9^2 - 2 \cdot 63 \cdot 41.9 \cdot \cos I \\ 180^\circ - 83.5^\circ - 55.1^\circ = 41.4^\circ\\ \angle{C} \approx 83.5^\circ\\ \angle{I} \approx 55.1^\circ \\ \angle{R} \approx 41.4^\circ \end{align*}

### Vocabulary

Law of Cosines: The law of cosines relates the lengths of the sides of a triangle that is not a right triangle.

### Guided Practice

1. Find AD\begin{align*}AD\end{align*} using the Pythagorean Theorem, Law of Cosines, trig functions, or any combination of the three.

2. Find HK\begin{align*}HK\end{align*} using the Pythagorean Theorem, Law of Cosines, trig functions, or any combination of the three if JK=3.6,KI=5.2,JI=1.9,HI=6.7\begin{align*}JK = 3.6, KI = 5.2, JI = 1.9, HI = 6.7\end{align*}, and KJI=96.3\begin{align*}\angle{KJI} = 96.3^\circ\end{align*}.

3. Use the Law of Cosines to determine whether or not the following triangle is drawn accurately. If not, determine how far the measurement of side "d" is from the correct value.

Solutions:

1. First, find AB\begin{align*}AB\end{align*}. AB2=14.22+152214.215cos37.4,AB=9.4.sin23.3=AD9.4,AD=3.7.\begin{align*}AB^2 = 14.2^2 + 15^2 - 2 \cdot 14.2 \cdot 15 \cdot \cos 37.4^\circ, AB = 9.4. \sin 23.3^\circ = \frac{AD}{9.4}, AD = 3.7.\end{align*}

2. HJI=18096.3=83.7\begin{align*}\angle{HJI} = 180^\circ - 96.3^\circ = 83.7^\circ\end{align*} (these two angles are a linear pair). 6.72=HJ2+1.922HJ1.9cos83.7\begin{align*}6.7^2 = HJ^2 + 1.9^2 - 2 \cdot HJ \cdot 1.9 \cdot \cos 83.7^\circ\end{align*}. This simplifies to the quadratic equation HJ20.417HJ41.28\begin{align*}HJ^2 - 0.417HJ - 41.28\end{align*}. Using the quadratic formula, we can determine that HJ6.64\begin{align*}HJ \approx 6.64\end{align*}. So, since HJ+JK=HK,6.64+3.6HK10.24\begin{align*}HJ + JK = HK, 6.64 + 3.6 \approx HK \approx 10.24\end{align*}.

3. To determine this, use the Law of Cosines and solve for d\begin{align*}d\end{align*} to determine if the picture is accurate. d2=122+24221224cos30,d=14.9\begin{align*}d^2 = 12^2 + 24^2 - 2 \cdot 12 \cdot 24 \cdot \cos 30^\circ, d = 14.9\end{align*}, which means d\begin{align*}d\end{align*} in the picture is off by 1.9.

### Concept Problem Solution

Since your friend is certain of the lengths of the sides of the triangle, you should use those as the known quantities in the Law of Cosines and solve for the angle:

72=52+42+(2)(5)(4)cosθ49=25+16+40cosθ492516=40cosθ840=cosθcos1840=θθ=78.46\begin{align*}7^2 = 5^2 + 4^2 + (2)(5)(4)\cos \theta\\ 49 = 25 + 16 + 40 \cos \theta\\ 49 - 25 - 16 = 40 \cos \theta\\ \frac{8}{40} = \cos \theta\\ cos^{-1} \frac{8}{40} = \theta\\ \theta = 78.46 \end{align*}

So as it turns out, your friend is rather close. Her measurements were probably slight inaccurate because of her round off from the protractor.

### Practice

1. If you know the lengths of all three sides of a triangle and the measure of one angle, how can you determine if the triangle is drawn accurately?

Determine whether or not each triangle is labelled correctly.

Determine whether or not each described triangle is possible. Assume angles have been rounded to the nearest degree.

1. In BCD\begin{align*}\triangle BCD\end{align*}, b=4, c=4, d=5, and mB=51\begin{align*}m\angle B=51^\circ\end{align*}.
2. In ABC\begin{align*}\triangle ABC\end{align*}, a=7, b=4, c=9, and mB=34\begin{align*}m\angle B=34^\circ\end{align*}.
3. In BCD\begin{align*}\triangle BCD\end{align*}, b=3, c=2, d=7, and mD=138\begin{align*}m\angle D=138^\circ\end{align*}.
4. In ABC\begin{align*}\triangle ABC\end{align*}, a=8, b=6, c=13.97, and mC=172\begin{align*}m\angle C=172^\circ\end{align*}.
5. In ABC\begin{align*}\triangle ABC\end{align*}, a=4, b=4, c=9, and \begin{align*}m\angle B=170^\circ\end{align*}.
6. In \begin{align*}\triangle BCD\end{align*}, b=3, c=5, d=4, and \begin{align*}m\angle C=90^\circ\end{align*}.
7. In \begin{align*}\triangle ABC\end{align*}, a=8, b=3, c=6, and \begin{align*}m\angle A=122^\circ\end{align*}.
8. If you use the Law of Cosines to solve for \begin{align*}m\angle C\end{align*} in \begin{align*}\triangle ABC\end{align*} where a=3, b=7, and c=12, you will an error. Explain why.

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### Vocabulary Language: English

law of cosines

The law of cosines is a rule relating the sides of a triangle to the cosine of one of its angles. The law of cosines states that $c^2=a^2+b^2-2ab\cos C$, where $C$ is the angle across from side $c$.

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