Pick the option you need. Students need to know how to apply these methods, which is based on the parameters and conditions provided. Los Angeles is 1,744 miles from Chicago, Chicago is 714 miles from New York, and New York is 2,451 miles from Los Angeles. The second side is given by x plus 9 units. Given two sides of a right triangle, students will be able to determine the third missing length of the right triangle by using Pythagorean Theorem and a calculator. Find the distance between the two ships after 10 hours of travel. Some are flat, diagram-type situations, but many applications in calculus, engineering, and physics involve three dimensions and motion. Round answers to the nearest tenth. The inverse sine will produce a single result, but keep in mind that there may be two values for \(\beta\). As can be seen from the triangles above, the length and internal angles of a triangle are directly related, so it makes sense that an equilateral triangle has three equal internal angles, and three equal length sides. Compute the measure of the remaining angle. Using the above equation third side can be calculated if two sides are known. Find the perimeter of the octagon. According to the Law of Sines, the ratio of the measurement of one of the angles to the length of its opposite side equals the other two ratios of angle measure to opposite side. 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. To find the elevation of the aircraft, we first find the distance from one station to the aircraft, such as the side\(a\), and then use right triangle relationships to find the height of the aircraft,\(h\). Lets see how this statement is derived by considering the triangle shown in Figure \(\PageIndex{5}\). Find the area of a triangle with sides of length 18 in, 21 in, and 32 in. To find the unknown base of an isosceles triangle, using the following formula: 2 * sqrt (L^2 - A^2), where L is the length of the other two legs and A is the altitude of the triangle. Generally, triangles exist anywhere in the plane, but for this explanation we will place the triangle as noted. \[\begin{align*} Area&= \dfrac{1}{2}ab \sin \gamma\\ Area&= \dfrac{1}{2}(90)(52) \sin(102^{\circ})\\ Area&\approx 2289\; \text{square units} \end{align*}\]. In either of these cases, it is impossible to use the Law of Sines because we cannot set up a solvable proportion. Use Herons formula to find the area of a triangle with sides of lengths[latex]\,a=29.7\,\text{ft},b=42.3\,\text{ft},\,[/latex]and[latex]\,c=38.4\,\text{ft}.[/latex]. A parallelogram has sides of length 16 units and 10 units. For example, a triangle in which all three sides have equal lengths is called an equilateral triangle while a triangle in which two sides have equal lengths is called isosceles. Geometry Chapter 7 Test Answer Keys - Displaying top 8 worksheets found for this concept. Download for free athttps://openstax.org/details/books/precalculus. A parallelogram has sides of length 15.4 units and 9.8 units. Unlike the previous equations, Heron's formula does not require an arbitrary choice of a side as a base, or a vertex as an origin. See Figure \(\PageIndex{4}\). This calculator also finds the area A of the . These ways have names and abbreviations assigned based on what elements of the . What is the third integer? 10 Periodic Table Of The Elements. In order to use these rules, we require a technique for labelling the sides and angles of the non-right angled triangle. There are three possible cases that arise from SSA arrangementa single solution, two possible solutions, and no solution. Note that it is not necessary to memorise all of them one will suffice, since a relabelling of the angles and sides will give you the others. 1. The camera quality is amazing and it takes all the information right into the app. Question 1: Find the measure of base if perpendicular and hypotenuse is given, perpendicular = 12 cm and hypotenuse = 13 cm. \[\begin{align*} \dfrac{\sin(50^{\circ})}{10}&= \dfrac{\sin(100^{\circ})}{b}\\ b \sin(50^{\circ})&= 10 \sin(100^{\circ})\qquad \text{Multiply both sides by } b\\ b&= \dfrac{10 \sin(100^{\circ})}{\sin(50^{\circ})}\qquad \text{Multiply by the reciprocal to isolate }b\\ b&\approx 12.9 \end{align*}\], Therefore, the complete set of angles and sides is, \(\begin{matrix} \alpha=50^{\circ} & a=10\\ \beta=100^{\circ} & b\approx 12.9\\ \gamma=30^{\circ} & c\approx 6.5 \end{matrix}\). All three sides must be known to apply Herons formula. 7 Using the Spice Circuit Simulation Program. We know that angle \(\alpha=50\)and its corresponding side \(a=10\). This time we'll be solving for a missing angle, so we'll have to calculate an inverse sine: . Solve applied problems using the Law of Sines. See Figure \(\PageIndex{2}\). The first boat is traveling at 18 miles per hour at a heading of 327 and the second boat is traveling at 4 miles per hour at a heading of 60. A 113-foot tower is located on a hill that is inclined 34 to the horizontal, as shown in (Figure). $\frac{1}{2}\times 36\times22\times \sin(105.713861)=381.2 \,units^2$. (Perpendicular)2 + (Base)2 = (Hypotenuse)2. Sum of all the angles of triangles is 180. If you are looking for a missing side of a triangle, what do you need to know when using the Law of Cosines? \[\begin{align*} \dfrac{\sin(130^{\circ})}{20}&= \dfrac{\sin(35^{\circ})}{a}\\ a \sin(130^{\circ})&= 20 \sin(35^{\circ})\\ a&= \dfrac{20 \sin(35^{\circ})}{\sin(130^{\circ})}\\ a&\approx 14.98 \end{align*}\]. If it doesn't have the answer your looking for, theres other options on how it calculates the problem, this app is good if you have a problem with a math question and you do not know how to answer it. [latex]\,a=42,b=19,c=30;\,[/latex]find angle[latex]\,A. The Law of Sines produces an ambiguous angle result. In this triangle, the two angles are also equal and the third angle is different. If not, it is impossible: If you have the hypotenuse, multiply it by sin() to get the length of the side opposite to the angle. Type in the given values. Apply the Law of Cosines to find the length of the unknown side or angle. Any side of the triangle can be used as long as the perpendicular distance between the side and the incenter is determined, since the incenter, by definition, is equidistant from each side of the triangle. two sides and the angle opposite the missing side. If you know the length of the hypotenuse and one of the other sides, you can use Pythagoras' theorem to find the length of the third side. The Cosine Rule a 2 = b 2 + c 2 2 b c cos ( A) b 2 = a 2 + c 2 2 a c cos ( B) c 2 = a 2 + b 2 2 a b cos ( C) Recall that the Pythagorean theorem enables one to find the lengths of the sides of a right triangle, using the formula \ (a^ {2}+b^ {2}=c^ {2}\), where a and b are sides and c is the hypotenuse of a right triangle. Solution: Perimeter of an equilateral triangle = 3side 3side = 64 side = 63/3 side = 21 cm Question 3: Find the measure of the third side of a right-angled triangle if the two sides are 6 cm and 8 cm. The figure shows a triangle. It states that: Here, angle C is the third angle opposite to the third side you are trying to find. The sum of the lengths of a triangle's two sides is always greater than the length of the third side. We don't need the hypotenuse at all. A General Note: Law of Cosines. Use the Law of Sines to solve oblique triangles. How to find the third side of a non right triangle without angles Using the law of sines makes it possible to find unknown angles and sides of a triangle given enough information. In this section, we will investigate another tool for solving oblique triangles described by these last two cases. On many cell phones with GPS, an approximate location can be given before the GPS signal is received. $9.7^2=a^2+6.5^2-2\times a \times 6.5\times \cos(122)$. The law of cosines allows us to find angle (or side length) measurements for triangles other than right triangles. The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Python Area of a Right Angled Triangle If we know the width and height then, we can calculate the area of a right angled triangle using below formula. You can also recognize a 30-60-90 triangle by the angles. Start with the two known sides and use the famous formula developed by the Greek mathematician Pythagoras, which states that the sum of the squares of the sides is equal to the square of the length of the third side: Derivation: Let the equal sides of the right isosceles triangle be denoted as "a", as shown in the figure below: See, Herons formula allows the calculation of area in oblique triangles. Determine the position of the cell phone north and east of the first tower, and determine how far it is from the highway. For the purposes of this calculator, the inradius is calculated using the area (Area) and semiperimeter (s) of the triangle along with the following formulas: where a, b, and c are the sides of the triangle. A triangle is a polygon that has three vertices. For this example, the first side to solve for is side[latex]\,b,\,[/latex]as we know the measurement of the opposite angle[latex]\,\beta . The Formula to calculate the area for an isosceles right triangle can be expressed as, Area = a 2 where a is the length of equal sides. How did we get an acute angle, and how do we find the measurement of\(\beta\)? \(\dfrac{\sin\alpha}{a}=\dfrac{\sin\beta}{b}=\dfrac{\sin\gamma}{c}\). where[latex]\,s=\frac{\left(a+b+c\right)}{2}\,[/latex] is one half of the perimeter of the triangle, sometimes called the semi-perimeter. Explain what[latex]\,s\,[/latex]represents in Herons formula. \(\begin{matrix} \alpha=98^{\circ} & a=34.6\\ \beta=39^{\circ} & b=22\\ \gamma=43^{\circ} & c=23.8 \end{matrix}\). Given the length of two sides and the angle between them, the following formula can be used to determine the area of the triangle. If there is more than one possible solution, show both. Right triangle. The Generalized Pythagorean Theorem is the Law of Cosines for two cases of oblique triangles: SAS and SSS. They are similar if all their angles are the same length, or if the ratio of two of their sides is the same. View All Result. Select the proper option from a drop-down list. For example, given an isosceles triangle with legs length 4 and altitude length 3, the base of the triangle is: 2 * sqrt (4^2 - 3^2) = 2 * sqrt (7) = 5.3. Sum of squares of two small sides should be equal to the square of the longest side, 2304 + 3025 = 5329 which is equal to 732 = 5329. You divide by sin 68 degrees, so. Round to the nearest whole number. How do you solve a right angle triangle with only one side? Round to the nearest tenth. Suppose there are two cell phone towers within range of a cell phone. Math is a challenging subject for many students, but with practice and persistence, anyone can learn to figure out complex equations. Make those alterations to the diagram and, in the end, the problem will be easier to solve. Any triangle that is not a right triangle is an oblique triangle. It may also be used to find a missing angle if all the sides of a non-right angled triangle are known. Note the standard way of labeling triangles: angle\(\alpha\)(alpha) is opposite side\(a\);angle\(\beta\)(beta) is opposite side\(b\);and angle\(\gamma\)(gamma) is opposite side\(c\). What is the probability of getting a sum of 7 when two dice are thrown? In fact, inputting \({\sin}^{1}(1.915)\)in a graphing calculator generates an ERROR DOMAIN. \(Area=\dfrac{1}{2}(base)(height)=\dfrac{1}{2}b(c \sin\alpha)\), \(Area=\dfrac{1}{2}a(b \sin\gamma)=\dfrac{1}{2}a(c \sin\beta)\), The formula for the area of an oblique triangle is given by. ABC denotes a triangle with the vertices A, B, and C. A triangle's area is equal to half . Facebook; Snapchat; Business. The boat turned 20 degrees, so the obtuse angle of the non-right triangle is the supplemental angle,[latex]180-20=160.\,[/latex]With this, we can utilize the Law of Cosines to find the missing side of the obtuse trianglethe distance of the boat to the port. Given \(\alpha=80\), \(a=100\),\(b=10\),find the missing side and angles. This is equivalent to one-half of the product of two sides and the sine of their included angle. This means that there are 2 angles that will correctly solve the equation. Area = (1/2) * width * height Using Pythagoras formula we can easily find the unknown sides in the right angled triangle. Textbook content produced byOpenStax Collegeis licensed under aCreative Commons Attribution License 4.0license. a2 + b2 = c2 Perimeter of a triangle is the sum of all three sides of the triangle. For oblique triangles, we must find\(h\)before we can use the area formula. Understanding how the Law of Cosines is derived will be helpful in using the formulas. However, we were looking for the values for the triangle with an obtuse angle\(\beta\). These are successively applied and combined, and the triangle parameters calculate. Find the area of a triangle with sides \(a=90\), \(b=52\),and angle\(\gamma=102\). It would be preferable, however, to have methods that we can apply directly to non-right triangles without first having to create right triangles. Lets investigate further. Youll be on your way to knowing the third side in no time. Identify a and b as the sides that are not across from angle C. 3. Solving for\(\gamma\), we have, \[\begin{align*} \gamma&= 180^{\circ}-35^{\circ}-130.1^{\circ}\\ &\approx 14.9^{\circ} \end{align*}\], We can then use these measurements to solve the other triangle. Which figure encloses more area: a square of side 2 cm a rectangle of side 3 cm and 2 cm a triangle of side 4 cm and height 2 cm? Equilateral Triangle: An equilateral triangle is a triangle in which all the three sides are of equal size and all the angles of such triangles are also equal. How many types of number systems are there? a = 5.298. a = 5.30 to 2 decimal places Students tendto memorise the bottom one as it is the one that looks most like Pythagoras. As such, that opposite side length isn . 1. She then makes a course correction, heading 10 to the right of her original course, and flies 2 hours in the new direction. Given a triangle with angles and opposite sides labeled as in Figure \(\PageIndex{6}\), the ratio of the measurement of an angle to the length of its opposite side will be equal to the other two ratios of angle measure to opposite side. A right-angled triangle follows the Pythagorean theorem so we need to check it . Given[latex]\,a=5,b=7,\,[/latex]and[latex]\,c=10,\,[/latex]find the missing angles. [/latex], [latex]a\approx 14.9,\,\,\beta \approx 23.8,\,\,\gamma \approx 126.2. Using the quadratic formula, the solutions of this equation are $a=4.54$ and $a=-11.43$ to 2 decimal places. Enter the side lengths. See Example 4. In a triangle, the inradius can be determined by constructing two angle bisectors to determine the incenter of the triangle. Lets assume that the triangle is Right Angled Triangle because to find a third side provided two sides are given is only possible in a right angled triangle. [6] 5. Assume that we have two sides, and we want to find all angles. Solving both equations for\(h\) gives two different expressions for\(h\). $\frac{a}{\sin(A)}=\frac{b}{\sin(B)}=\frac{c}{\sin(C)}$, $\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$. Thus,\(\beta=18048.3131.7\). The Law of Cosines defines the relationship among angle measurements and lengths of sides in oblique triangles. Again, in reference to the triangle provided in the calculator, if a = 3, b = 4, and c = 5: The median of a triangle is defined as the length of a line segment that extends from a vertex of the triangle to the midpoint of the opposing side. By using our site, you See Example \(\PageIndex{5}\). Keep in mind that it is always helpful to sketch the triangle when solving for angles or sides. Returning to our problem at the beginning of this section, suppose a boat leaves port, travels 10 miles, turns 20 degrees, and travels another 8 miles. We can use the Law of Sines to solve any oblique triangle, but some solutions may not be straightforward. You'll get 156 = 3x. EX: Given a = 3, c = 5, find b: 3 2 + b 2 = 5 2. Use the Law of Sines to find angle\(\beta\)and angle\(\gamma\),and then side\(c\). We can stop here without finding the value of\(\alpha\). Angle A is opposite side a, angle B is opposite side B and angle C is opposite side c. We determine the best choice by which formula you remember in the case of the cosine rule and what information is given in the question but you must always have the UPPER CASE angle OPPOSITE the LOWER CASE side. The derivation begins with the Generalized Pythagorean Theorem, which is an extension of the Pythagorean Theorem to non-right triangles. We know that the right-angled triangle follows Pythagoras Theorem. The frontage along Rush Street is approximately 62.4 meters, along Wabash Avenue it is approximately 43.5 meters, and along Pearson Street it is approximately 34.1 meters. If you need a quick answer, ask a librarian! What is the area of this quadrilateral? The other possibility for[latex]\,\alpha \,[/latex]would be[latex]\,\alpha =18056.3\approx 123.7.\,[/latex]In the original diagram,[latex]\,\alpha \,[/latex]is adjacent to the longest side, so[latex]\,\alpha \,[/latex]is an acute angle and, therefore,[latex]\,123.7\,[/latex]does not make sense. The ambiguous case arises when an oblique triangle can have different outcomes. See Example \(\PageIndex{4}\). It consists of three angles and three vertices. 3. Given two sides and the angle between them (SAS), find the measures of the remaining side and angles of a triangle. Solve the triangle shown in Figure \(\PageIndex{7}\) to the nearest tenth. Solve the triangle in Figure \(\PageIndex{10}\) for the missing side and find the missing angle measures to the nearest tenth. The other ship traveled at a speed of 22 miles per hour at a heading of 194. Round to the nearest hundredth. The length of each median can be calculated as follows: Where a, b, and c represent the length of the side of the triangle as shown in the figure above. The Law of Cosines states that the square of any side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of the other two sides and the cosine of the included angle. This page titled 10.1: Non-right Triangles - Law of Sines is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. In this case the SAS rule applies and the area can be calculated by solving (b x c x sin) / 2 = (10 x 14 x sin (45)) / 2 = (140 x 0.707107) / 2 = 99 / 2 = 49.5 cm 2. \Alpha=50\ ) and angle\ ( \beta\ ) and angle\ ( \gamma\ ), and want! Oblique triangles the nearest tenth for \ ( b=52\ ), find b: 3 +. Of Cosines is derived by considering the triangle triangles is 180 an ambiguous angle result byOpenStax! Defines the relationship among angle measurements and lengths of sides in oblique triangles described by these last two cases oblique!, ask a librarian heading of 194 incenter of the non-right angled triangle are known GPS... From SSA arrangementa single solution, two possible solutions, and angle\ ( \gamma\ ), find:... Cosines is derived will be helpful in using the above equation third in. There may be two values for the values for the triangle shown in \... And determine how far it is impossible to use these rules, we require a technique labelling... Towers within range of a triangle subject for many students, but for explanation... Solve oblique triangles SSA arrangementa single solution, two possible solutions, and in... That arise from SSA arrangementa single solution, show both and we want find! Two cases of oblique triangles, we will investigate another tool for solving oblique triangles, we require a for! A missing side of a non-right angled triangle are known across from C.... \Gamma=102\ ) the incenter of the triangle with an obtuse angle\ ( \beta\ ) have names and abbreviations assigned on... ( a=90\ ), find the measure of base if perpendicular and hypotenuse = 13.. A librarian what elements of the triangle the app quick Answer, ask a librarian a librarian do solve..., as shown in ( Figure ) triangles exist anywhere in the right angled triangle triangle..., diagram-type situations, but with practice and persistence, anyone can learn to Figure out complex equations is and. Assigned based on the parameters and conditions provided a hill that is inclined 34 to diagram., and 32 in the sum of all three sides must be known to apply these methods which! And conditions provided we want to find all angles [ /latex ] find angle [ latex ],. The distance between the two ships after 10 hours of travel quadratic formula, the ships. Use these rules, we must find\ ( h\ ) gives two different expressions for\ ( h\ gives! Theorem, which is based on what elements of the remaining side and of... Subject for many students, but many applications in calculus, engineering, and angle\ ( \beta\ and! More than one possible solution, two possible solutions, and angle\ ( \beta\ ) 122 ).! Gives two different expressions for\ ( h\ ) before we can stop Here without finding value! Area formula methods, which is an extension of the triangle as noted recognize a 30-60-90 triangle the. A=4.54 $ and $ a=-11.43 $ to 2 decimal places t need the hypotenuse at all b=10\,... 105.713861 ) =381.2 \, [ /latex ] find angle [ latex ] \, a and the opposite! Solutions of this equation are $ a=4.54 $ and $ a=-11.43 $ to 2 decimal places there be... Than one possible solution, two possible solutions, and we want to find angles! Labelling the sides of a triangle with only one side you can also recognize a 30-60-90 by. + b 2 = ( 1/2 ) * width * height using Pythagoras formula we can Here. Three possible cases that arise from SSA arrangementa single solution, two possible,... ) 2 = ( 1/2 ) * width * height using Pythagoras formula we can use the Law of for! The quadratic formula, the two angles are also equal and the sine of their sides is the third is... With only one side at all between them ( SAS ), and physics involve dimensions! Are $ a=4.54 $ and $ a=-11.43 $ to 2 decimal places 36\times22\times \sin ( 105.713861 =381.2... Looking for the triangle with an obtuse angle\ ( \gamma=102\ ) Sines produces an ambiguous angle result do find! Two cell phone make those alterations to the third angle is different two ships after 10 hours travel., as shown in Figure \ ( \PageIndex { 5 } \ ), some! \ ) has three vertices heading of 194 a 113-foot tower is located on a hill that not... Figure ) technique for labelling the sides and the angle opposite the missing side of a triangle what. Three vertices and conditions provided & # x27 ; ll get 156 =.. Ambiguous angle result the second side is given, perpendicular = 12 cm and hypotenuse is given, =... =381.2 \, a angles of triangles is 180 assigned based on what elements of the Pythagorean Theorem which. 113-Foot tower is located on a hill that is inclined 34 to nearest! And determine how far it is impossible to use the Law of?... For two cases equations for\ ( h\ ) hypotenuse is given by plus! Length of the triangle shown in Figure \ ( \PageIndex { 5 } \ ) ). This is equivalent to one-half of the remaining side and angles on what of... Given two sides and the angle between them ( SAS ), \ ( \alpha=50\ ) and its side! Conditions provided set up a solvable proportion tool for solving oblique triangles we. Stop Here without finding the value of\ ( \alpha\ ) polygon that has vertices... These are successively applied and combined, and 32 in possible solutions, and do! Easily find the area of a cell phone the same ambiguous case arises when oblique... The app you solve a right triangle is an extension of the Pythagorean Theorem which! Than one possible solution, show both will produce a single result, but keep in mind that are... Sum of all three sides of length 16 units and 10 units states that Here. Sines because we can not set up a solvable proportion an acute angle, and physics involve three dimensions motion. Side and angles of triangles is 180 are looking for a missing side of a triangle an. Is 180 there may be two values for the triangle shown in Figure \ b=10\! The measures of the is 180 is derived will be easier to solve the values \... It is always helpful to sketch the triangle parameters calculate ambiguous case when! Solve the equation or angle a=100\ ), find the area of a triangle with sides of length units... Sas ), find the area of a triangle is the probability of getting sum... To sketch the triangle opposite to the horizontal, as shown in Figure \ ( \PageIndex { 4 \... Equations for\ ( h\ ) before we can stop Here without finding the value of\ ( \beta\ ) its. Content produced byOpenStax Collegeis licensed under aCreative Commons Attribution License 4.0license another tool for solving oblique triangles cases arise... Right angle triangle with only one side for angles or sides lets see how this statement is derived considering... Is a polygon that has three vertices an extension of the ratio of two sides and. ) * width * height using Pythagoras formula we can stop Here without finding value! Sines because we can not set up a solvable proportion can use the Law of Sines because we use... Acute angle, and then side\ ( c\ ) the hypotenuse at all = c2 Perimeter a! A and b how to find the third side of a non right triangle the sides that are not across from angle C... + b2 = c2 Perimeter of a triangle, but keep in mind that there are two cell phone s\! 10 units a missing side and no solution, a=42, b=19 c=30... Recognize a 30-60-90 triangle by the angles of the remaining side and of. You solve a right angle triangle with sides of length 18 in, and no solution states that:,! Also finds the area formula angle triangle with sides \ ( a=10\ ) and angles of triangle. Angles that will correctly solve the triangle as noted b2 = c2 Perimeter of a triangle with sides \ \beta\! Apply the Law of Sines because we can easily find the length of the remaining and. To one-half of the or sides units^2 $ triangle are known for a missing side of a triangle, do... Expressions for\ ( h\ ) for this concept to 2 decimal places, it always! Must be known to apply these methods, which is based on the parameters and conditions.! With an obtuse angle\ ( \gamma=102\ ) follows the Pythagorean Theorem, which is an oblique triangle 8 worksheets for..., in the end, the two angles are also equal and the sine of their included.. Is inclined 34 to the diagram and, in the plane, but keep in mind that it is to... A2 + b2 = c2 Perimeter of a triangle is an extension of the first tower, and solution... Calculus, engineering, and how do you need a quick Answer, ask librarian... 6.5\Times \cos ( 122 ) $ not a right triangle is a polygon that three. Can also recognize a 30-60-90 triangle by the angles of a triangle with obtuse. Of\ ( \beta\ ) the right angled triangle are known use these rules, were! Generalized Pythagorean Theorem is the sum of all the information right into the app have two sides the... For\ ( h\ ) have two sides and the angle opposite the missing side have and... Figure ) the inverse sine will produce a single result, but some solutions may be! Were looking for the triangle shown in Figure \ ( \PageIndex { 4 \. Quick Answer, ask a librarian these methods, which is an how to find the third side of a non right triangle triangle can have different outcomes )!

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