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9.6 Solving Right
Triangles
Geometry
Mrs. Spitz
Spring 2005
Objectives/Assignment
• Solve a right triangle.
• Use right triangles to solve real-life
problems, such as finding the glide angle
and altitude of a space shuttle.
• To solve problems like determining the
correct dimensions of a wheel-chair ramp.
• Assignment: pp. 570-571 #4-38
• Due today 9.5 and Quiz at the end of
class.
Solving a right triangle
• Every right triangle has one right angle,
two acute angles, one hypotenuse and two
legs. To solve a right triangle, means to
determine the measures of all six (6) parts.
You can solve a right triangle if the
following one of the two situations exist:
– Two side lengths
– One side length and one acute angle
measure
Note:
• As you learned in Lesson 9.5, you can use
the side lengths of a right triangle to find
trigonometric ratios for the acute angles of
the triangle. As you will see in this lesson,
once you know the sine, cosine, or tangent
of an acute angle, you can use a
calculator to find the measure of the angle.
WRITE THIS DOWN!!!
• In general, for an acute angle A:
– If sin A = x, then sin-1 x = mA
– If cos A = y, then cos-1 y = mA
– If tan A = z, then tan-1 z = mA
The expression sin-1 x is read as “the inverse sine
of x.”
• On your calculator, this means you will be
punching the 2nd function button usually in
yellow prior to doing the calculation. This is to
find the degree of the angle.
Example 1:
• Solve the right
triangle. Round the
decimals to the
nearest tenth.
c
3 2
B
C
A
HINT: Start by using the Pythagorean Theorem.
You have side a and side b. You don’t have the
hypotenuse which is side c—directly across from
the right angle.
Example 1:
c
3 2
B
C
A
(hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem
c2 = 32 + 22 Substitute values
c2 = 9 + 4 Simplify
Simplify
c = √13
c2 = 13
Find the positive square root
c ≈ 3.6 Use a calculator to approximate
Example 1 continued
• Then use a calculator to find the measure
of B:
2nd function
Tangent button
2
Divided by symbol
3 ≈ 33.7°
Finally
• Because A and B are complements,
you can write
mA = 90° - mB ≈ 90° - 33.7° = 56.3°
The side lengths of the triangle are 2, 3
and √13, or about 3.6. The triangle has
one right angle and two acute angles
whose measure are about 33.7° and
56.3°.
Ex. 2: Solving a Right Triangle (h)
• Solve the right
triangle. Round
decimals to the
nearest tenth.
13 h
g
H J
G
25°
You are looking for
opposite and
hypotenuse which is
the sin ratio.
sin H =
opp.
hyp.
13 sin 25° =
h
13
13
13(0.4226) ≈ h
5.5 ≈ h
Set up the correct ratio
Substitute values/multiply by reciprocal
Substitute value from table or calculator
Use your calculator to approximate.
Ex. 2: Solving a Right Triangle (g)
• Solve the right
triangle. Round
decimals to the
nearest tenth.
13 h
g
H J
G
25°
You are looking for
adjacent and
hypotenuse which is
the cosine ratio.
cos G =
adj.
hyp.
13 cos 25° =
g
13
13
13(0.9063) ≈ g
11.8 ≈ h
Set up the correct ratio
Substitute values/multiply by reciprocal
Substitute value from table or calculator
Use your calculator to approximate.
Using Right Triangles in Real Life
• Space Shuttle: During its
approach to Earth, the
space shuttle’s glide
angle changes.
• A. When the shuttle’s
altitude is about 15.7
miles, its horizontal
distance to the runway is
about 59 miles. What is
its glide angle? Round
your answer to the
nearest tenth.
Solution:
• You know opposite
and adjacent sides. If
you take the opposite
and divide it by the
adjacent sides, then
take the inverse
tangent of the ratio,
this will yield you the
slide angle.
distance to runway
altitude
Glide  = x°
59 miles
15.7
miles
tan x° =
opp.
adj.
tan x° =
15.7
59
Key in calculator 2nd function,
tan 15.7/59 ≈ 14.9
Use correct ratio
Substitute values
 When the space shuttle’s altitude is about 15.7 miles, the
glide angle is about 14.9°.
B. Solution
• When the space
shuttle is 5 miles from
the runway, its glide
angle is about 19°.
Find the shuttle’s
altitude at this point in
its descent. Round
your answer to the
nearest tenth.
distance to runway
altitude
Glide  = 19°
5 miles
h
tan 19° =
opp.
adj.
tan 19° =
h
5
Use correct ratio
Substitute values
5 tan 19° =
h
5
5 Isolate h by
multiplying by 5.
1.7 ≈ h Approximate using calculator
 The shuttle’s altitude is
about 1.7 miles.
Next week: 2nd, 4th and 6th Period
• 9.7: Vectors – Wednesday
• Ch. 9 Review – Friday
• Ch. 9 Test/Binder Check – Monday before
you go on Spring Break – Make sure you
take this if you are not planning at being in
school before this date.
• Chapter 10 Post/Thms. And Ch. 10
Definitions – Due Wednesday
• 10.1 Friday before you leave.
Next week: 5th period only
• 9.7: Vectors -- Friday
• HW: Chapter 9 Review – Bring to class
completed on Monday.
• Ch. 9 Test/Binder Check – Monday before
you go on Spring Break – Make sure you
take this if you are not planning at being in
school before this date.
• Baskin & Robbins -- Wednesday

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Solving The Right Triangles PowerPoint 2.ppt

  • 2. Objectives/Assignment • Solve a right triangle. • Use right triangles to solve real-life problems, such as finding the glide angle and altitude of a space shuttle. • To solve problems like determining the correct dimensions of a wheel-chair ramp. • Assignment: pp. 570-571 #4-38 • Due today 9.5 and Quiz at the end of class.
  • 3. Solving a right triangle • Every right triangle has one right angle, two acute angles, one hypotenuse and two legs. To solve a right triangle, means to determine the measures of all six (6) parts. You can solve a right triangle if the following one of the two situations exist: – Two side lengths – One side length and one acute angle measure
  • 4. Note: • As you learned in Lesson 9.5, you can use the side lengths of a right triangle to find trigonometric ratios for the acute angles of the triangle. As you will see in this lesson, once you know the sine, cosine, or tangent of an acute angle, you can use a calculator to find the measure of the angle.
  • 5. WRITE THIS DOWN!!! • In general, for an acute angle A: – If sin A = x, then sin-1 x = mA – If cos A = y, then cos-1 y = mA – If tan A = z, then tan-1 z = mA The expression sin-1 x is read as “the inverse sine of x.” • On your calculator, this means you will be punching the 2nd function button usually in yellow prior to doing the calculation. This is to find the degree of the angle.
  • 6. Example 1: • Solve the right triangle. Round the decimals to the nearest tenth. c 3 2 B C A HINT: Start by using the Pythagorean Theorem. You have side a and side b. You don’t have the hypotenuse which is side c—directly across from the right angle.
  • 7. Example 1: c 3 2 B C A (hypotenuse)2 = (leg)2 + (leg)2 Pythagorean Theorem c2 = 32 + 22 Substitute values c2 = 9 + 4 Simplify Simplify c = √13 c2 = 13 Find the positive square root c ≈ 3.6 Use a calculator to approximate
  • 8. Example 1 continued • Then use a calculator to find the measure of B: 2nd function Tangent button 2 Divided by symbol 3 ≈ 33.7°
  • 9. Finally • Because A and B are complements, you can write mA = 90° - mB ≈ 90° - 33.7° = 56.3° The side lengths of the triangle are 2, 3 and √13, or about 3.6. The triangle has one right angle and two acute angles whose measure are about 33.7° and 56.3°.
  • 10. Ex. 2: Solving a Right Triangle (h) • Solve the right triangle. Round decimals to the nearest tenth. 13 h g H J G 25° You are looking for opposite and hypotenuse which is the sin ratio. sin H = opp. hyp. 13 sin 25° = h 13 13 13(0.4226) ≈ h 5.5 ≈ h Set up the correct ratio Substitute values/multiply by reciprocal Substitute value from table or calculator Use your calculator to approximate.
  • 11. Ex. 2: Solving a Right Triangle (g) • Solve the right triangle. Round decimals to the nearest tenth. 13 h g H J G 25° You are looking for adjacent and hypotenuse which is the cosine ratio. cos G = adj. hyp. 13 cos 25° = g 13 13 13(0.9063) ≈ g 11.8 ≈ h Set up the correct ratio Substitute values/multiply by reciprocal Substitute value from table or calculator Use your calculator to approximate.
  • 12. Using Right Triangles in Real Life • Space Shuttle: During its approach to Earth, the space shuttle’s glide angle changes. • A. When the shuttle’s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.
  • 13. Solution: • You know opposite and adjacent sides. If you take the opposite and divide it by the adjacent sides, then take the inverse tangent of the ratio, this will yield you the slide angle. distance to runway altitude Glide  = x° 59 miles 15.7 miles tan x° = opp. adj. tan x° = 15.7 59 Key in calculator 2nd function, tan 15.7/59 ≈ 14.9 Use correct ratio Substitute values  When the space shuttle’s altitude is about 15.7 miles, the glide angle is about 14.9°.
  • 14. B. Solution • When the space shuttle is 5 miles from the runway, its glide angle is about 19°. Find the shuttle’s altitude at this point in its descent. Round your answer to the nearest tenth. distance to runway altitude Glide  = 19° 5 miles h tan 19° = opp. adj. tan 19° = h 5 Use correct ratio Substitute values 5 tan 19° = h 5 5 Isolate h by multiplying by 5. 1.7 ≈ h Approximate using calculator  The shuttle’s altitude is about 1.7 miles.
  • 15. Next week: 2nd, 4th and 6th Period • 9.7: Vectors – Wednesday • Ch. 9 Review – Friday • Ch. 9 Test/Binder Check – Monday before you go on Spring Break – Make sure you take this if you are not planning at being in school before this date. • Chapter 10 Post/Thms. And Ch. 10 Definitions – Due Wednesday • 10.1 Friday before you leave.
  • 16. Next week: 5th period only • 9.7: Vectors -- Friday • HW: Chapter 9 Review – Bring to class completed on Monday. • Ch. 9 Test/Binder Check – Monday before you go on Spring Break – Make sure you take this if you are not planning at being in school before this date. • Baskin & Robbins -- Wednesday