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Solving Right Triangles
Topic 1 Solving Right Triangles Unit 3 Topic 1
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Information Pythagorean Theorem
c2 a2 b2 a2 + b2 = c2 Information Pythagorean Theorem The sum of the areas of the squares attached to the legs of the triangle equals the area of the square attached to the hypotenuse. When two side lengths of a right triangle are known, we can use the Pythagorean Theorem, , to solve for the length of the missing side.
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Example 1a Try this on your own first!!!! Determining a Side Determine the length of the missing side in each triangle. Round to the nearest tenth. A B C 5 cm 8 cm
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Example 1a Solution Determining a Side Determine the length of the missing side in each triangle. Round to the nearest tenth. Solution: A B C 5 cm 8 cm Substitute known values into the formula, keeping in mind that the c value is reserved for the hypotenuse. Take the square root of both sides.
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Example 1b Try this on your own first!!!!
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Example 1b Solution Solution:
Substitute known values into the formula, keeping in mind that the hypotenuse (17) must be put into the formula as c. Subtract 64 from both sides. Take the square root of both sides.
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More Information Right ∆ABC is shown below. In relation to A, the sides are labelled as shown.
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Information SOH CAH TOA is the word that we use to remember what to do with right triangles to solve for either a missing angle (when given two sides), or a missing side (when given one angle and one side). A B C θ
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Information SOH CAH TOA is the word that we use to remember these ratios.
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Information We can use these with right triangles to solve for either a missing angle or a missing side. In order to solve for a missing value when given two values, we Label the sides as opposite, adjacent and hypotenuse. Determine which trig ratio (sine, cosine, or tangent) includes the two known measures and the one unknown measure. Substitute all known values into the chosen trig ratio equation, including the two known measures and the one unknown measure.
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Information Solve for the unknown value. If your missing measurement is a side length, algebraically determine. If your missing measurement is an angle, use the inverse function (sin-1, cos-1 or tan-1) to find the angle. When asked to solve a triangle, find all missing sides and angles.
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Example 2 Try this on your own first!!!! Solve a Right Triangle, Given Two Side Lengths Solve the triangle. Round all sides to the nearest tenth and all angles to the nearest degree. Helpful Hint The sum of the interior angles of any triangle is 180. In a right triangle, one angle is 90 and the sum of the two acute angles is 90.
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Example 2: Solution Solve a Right Triangle, Given Two Side Lengths Solve the triangle. Round all sides to the nearest tenth and all angles to the nearest degree. Label the hypotenuse. Use Pythagorean Theorem to solve for the missing side. b a c
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Example 2: Solution Continued
28.3 cm adjacent hypotenuse opposite Label the sides as opposite, adjacent and hypotenuse, with respect to the angle that you are solving for. Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side. Substitute all known values into the equation and solve. Solve for the remaining angle.
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Example 2: Solution Complete
28.3 cm Check that all angles and sides are solved for by completing the triangle.
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Example 3 Try this on your own first!!!! Solve a Right Triangle, Given One Side Length and One Angle Measure Solve the triangle. Round all sides to the nearest tenth and all angles to the nearest degree.
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Example 3: Solution Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle. Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side. Substitute all known values into the equation, including the variable for the unknown side length. hypotenuse c b adjacent a opposite
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Example 3: Solution ∠𝐴=70°, ∠𝐶=90° ∠𝐵+∠𝐶+∠𝐴=180° ∠𝐵=180°−90°−70° ∠𝐵=20°
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Example 3: Solution Complete
19.8 cm 54.5 cm Check that all angles and sides are solved for by completing the triangle.
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Example 4 Try this on your own first!!!! Solve a Right Triangle, Given No Picture Solve ΔPQR, where P = 46, R = 90, and r = 12 m. Round all sides to the nearest tenth and all angles to the nearest degree.
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Example 4: Solution Solve ΔPQR, where P = 46, R = 90, and r = 12 m. Round all sides to the nearest tenth and all angles to the nearest degree. P Q R 12 m Draw the diagram and label all information provided.
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Example 4: Solution P Q R 12 m hypotenuse adjacent opposite r q p
Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle. Determine which trig ratio (sin, cos, or tan) includes the given angle, the given side, and the missing side. Substitute all known values into the equation, including the variable for the unknown side length. P Q R 12 m hypotenuse adjacent opposite r q p
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Example 4: Solution P Q R 12 m ∠𝑃=46°, ∠𝑅=90° ∠𝑄+∠𝑅+∠𝑃=180° ∠Q=180°−90°−46° ∠Q=44°
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Example 4: Solution Complete
Q R 12 m 8.3 cm 8.6 cm Check that all angles and sides are solved for by completing the triangle.
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Need to Know: • When you are asked to solve a triangle, use known information to find all missing sides and angles. • In order to solve a right triangle, use the Pythagorean Theorem and the primary trigonometric ratios. • In ΔABC, the Pythagorean Theorem states • In ΔABC, the primary trig ratios are • An acronym for remembering the primary trig ratios is SOH CAH TOA. You’re ready! Try the homework from this section.
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