Topic 1 Solving Right Triangles Unit 3 Topic 1. Information Pythagorean Theorem The sum of the areas...

Topic 1Solving Right Triangles

Unit 3 Topic 1

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.

2 2 2 a b c

c2

a2

b2

a2 + b2 = c2

Example 1aDetermining a Side

Determine the length of the missing side in each triangle. Round to the nearest tenth.

Try this on your own first!!!!

A

BC

5 cm

8 cm

Substitute known values into the formula, keeping in mind that the c value is reserved for the hypotenuse.

Example 1a SolutionDetermining a Side

Determine the length of the missing side in each triangle. Round to the nearest tenth.

A

BC

5 cm

8 cm

Solution:

2 2 2

2 2 2

2

5 8

89

9.4cm

a b c

c

c

c

Take the square root of both

sides.

Example 1bTry this on your own first!!!!

Substitute known values into the formula, keeping in mind that the hypotenuse (17) must be put into the formula as c.

Example 1b Solution

Solution:2 2 2

2 2 2

2

2

2

8 17

64 289

225

15 cm

a b c

b

b

b

b

Take the square root of both sides.

Subtract 64 from both sides.

More Information

Right ∆ABC is shown below. In relation to A, the sides are labelled as shown.

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

BCθ

Information

• SOH CAH TOA is the word that we use to remember these ratios.

InformationWe 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, we1. Label the sides as opposite, adjacent and

hypotenuse.2. Determine which trig ratio (sine, cosine, or tangent)

includes the two known measures and the one unknown measure.

3. Substitute all known values into the chosen trig ratio equation, including the two known measures and the one unknown measure.

Information4. 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.

Example 2

Solve the triangle. Round all sides to the nearest tenth and all angles to the nearest degree.


Solve a Right Triangle, Given Two Side Lengths

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.

Example 2: Solution


Solve a Right Triangle, Given Two Side Lengths

Label the hypotenuse. Use Pythagorean Theorem to solve

for the missing side.

2 2 2

2 2 2

2

2

15 24

801

801

28.3 cm

a b c

c

c

c

c

b

c

a

Example 2: Solution Continued

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.

28.3 cm

adjacent

hypotenuse

opposite

1

3

Oppositetan

Adjacent

15tan

24

2 , 90

180

1

15tan

80 90 32

2

3

5

2

8

4

A C

B C A

B

B

A

A

A

Example 2: Solution Complete

Check that all angles and sides are solved for by completing the triangle.

28.3 cm

32

58

Example 3



Solve a Right Triangle, Given One Side Length and One Angle Measure

oppositesin( )

hypotenuse

sin(70 )58

58sin(70 )

54.5 cm

a

a

a

Example 3: Solution Label the sides as opposite, adjacent and hypotenuse, with respect to the given angle.


Substitute all known values into the equation, including the variable for the unknown side length.

opposite

hypotenuse

adjacent

a

b c

adjacentcos( )

hypotenuse

cos(70 )58

58cos(70 )

19.8 cm

b

b

b

Example 3: Solution

∠ 𝐴=70 ° , ∠𝐶=90 °∠𝐵+∠𝐶+∠𝐴=180 °∠𝐵=180 ° −90 ° −70 °

∠𝐵=20 °



2054.5 cm

19.8 cm

Example 4Solve 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.


Example 4: SolutionSolve ΔPQR, where P = 46, R = 90, and r = 12 m. Round all sides to the nearest tenth and all angles to the nearest degree.

Draw the diagram and label all information provided.

P

QR

4612 m

oppositesin( )

hypotenuse

sin(46 )12

12sin(46 )

8.6 cm

p

p

p

Example 4: Solution Label the sides as opposite,

adjacent and hypotenuse, with respect to the given angle.


Substitute all known values into the equation, including the variable for the unknown side length.

adjacentcos( )

hypotenuse

cos(46 )12

12cos(46 )

8.3 cm

q

q

q

P

QR

4612 m

hypotenuse

adjacent

opposite

rq

p

Example 4: Solution

∠𝑃=46 ° , ∠𝑅=90 °∠𝑄+∠𝑅+∠𝑃=180 °∠Q=180 ° −90 °−46 °

∠Q=44 °

P

QR

4612 m



448.6 cm

8.3 cm

P

QR

4612 m

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.

2 2 2a b c

You’re ready! Try the homework from this section.

Topic 1 Solving Right Triangles Unit 3 Topic 1. Information Pythagorean Theorem The sum of the areas...

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