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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.
Try this on your own first!!!!
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 the triangle. Round all sides to the nearest tenth and all angles to the nearest degree.
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 the triangle. Round all sides to the nearest tenth and all angles to the nearest degree.
Try this on your own first!!!!
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.
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.
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 °
Example 3: Solution Complete
Check that all angles and sides are solved for by completing the triangle.
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.
Try this on your own first!!!!
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.
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.
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
Example 4: Solution Complete
Check that all angles and sides are solved for by completing the triangle.
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.