Glencoe Geometry Interactive Chalkboard
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Lesson 7-1 Geometric Mean
Lesson 7-2 The Pythagorean Theorem and Its Converse
Lesson 7-3 Special Right Triangles
Lesson 7-4 Trigonometry
Lesson 7-5 Angles of Elevation and Depression
Lesson 7-6 The Law of Sines
Lesson 7-7 The Law of Cosines
Example 1 Geometric Mean
Example 2 Altitude and Segments of the Hypotenuse
Example 3 Altitude and Length of the Hypotenuse
Example 4 Hypotenuse and Segment of Hypotenuse
Find the geometric mean between 2 and 50.
Let x represent the geometric mean.
Definition of geometric mean
Cross products
Take the positive square
root of each side.
Simplify.
Answer: The geometric mean is 10.
Find the geometric mean between 25 and 7.
Let x represent the geometric mean.
Definition of geometric mean
Cross products
Take the positive square
root of each side.
Simplify.
Use a calculator.
Answer: The geometric mean is about 13.2.
a. Find the geometric mean between 3 and 12.
Answer: 6
b. Find the geometric mean between 4 and 20.
Answer: 8.9
Cross products
Take the positive square
root of each side.
Use a calculator.
Answer: CD is about 12.7.
Answer: about 8.5
KITES Ms. Alspach is constructing a kite for her son.
She has to arrange perpendicularly two support rods,
the shorter of which is 27 inches long. If she has to
place the short rod 7.25 inches from one end of the
long rod in order to form two right triangles with the
kite fabric, what is the length of the long rod?
Draw a diagram of one of the right triangles formed.
Let
be the altitude drawn from the right angle of
Cross products
Divide each side by 7.25.
Answer: The length of the long rod is 7.25 + 25.2, or
about 32.4 inches long.
AIRPLANES A jetliner has a wingspan, BD, of 211
feet. The segment drawn from the front of the plane to
the tail,
intersects
at point E. If AE is 163 feet,
what is the length of the aircraft?
Answer: about 231.3 ft
Find c and d in
is the altitude of right triangle JKL. Use Theorem 7.2
to write a proportion.
Cross products
Divide each side by 5.
is the leg of right triangle JKL. Use the Theorem 7.3 to
write a proportion.
Cross products
Take the square root.
Simplify.
Use a calculator.
Answer:
Find e and f.
f
Answer:
Example 1 Find the Length of the Hypotenuse
Example 2 Find the Length of a Leg
Example 3 Verify a Triangle is a Right Triangle
Example 4 Pythagorean Triples
LONGITUDE AND LATITUDE Carson City, Nevada, is
located at about 120 degrees longitude and 39
degrees latitude. NASA Ames is located about 122
degrees longitude and 37 degrees latitude. Use the
lines of longitude and latitude to find the degree
distance to the nearest tenth degree if you were to
travel directly from NASA Ames to Carson City.
The change in longitude between NASA Ames and Carson
City is
or 2 degrees. Let this distance be a.
The change in latitude is
Let this distance be b.
or 2 degrees latitude.
Use the Pythagorean Theorem to find the distance in
degrees from NASA Ames to Carson City, represented
by c.
Pythagorean Theorem
Simplify.
Add.
Take the square root of
each side.
Use a calculator.
Answer: The degree distance between NASA Ames
and Carson City is about 2.8 degrees.
LONGITUDE AND LATITUDE Carson City, Nevada, is
located at about 120 degrees longitude and 39
degrees latitude. NASA Dryden is located about 117
degrees longitude and 34 degrees latitude. Use the
lines of longitude and latitude to find the degree
distance to the nearest tenth degree if you were to
travel directly from NASA Dryden to Carson City.
Answer: about 5.8 degrees
Find d.
Pythagorean Theorem
Simplify.
Subtract 9 from each side.
Take the square root of
each side.
Use a calculator.
Answer:
Find x.
Answer:
COORDINATE GEOMETRY Verify that
triangle.
is a right
Use the Distance Formula to determine the lengths of the
sides.
Subtract.
Simplify.
Subtract.
Simplify.
Subtract.
Simplify.
By the converse of the Pythagorean Theorem, if the sum
of the squares of the measures of two sides of a triangle
equals the square of the measure of the longest side, then
the triangle is a right triangle.
Converse of the
Pythagorean Theorem
Simplify.
Add.
Answer: Since the sum of the squares of two sides
equals the square of the longest side,
is a right triangle.
COORDINATE GEOMETRY Verify that
triangle.
is a right
Answer:
is a
right triangle because
Determine whether 9, 12, and 5 are the sides of a right
triangle. Then state whether they form a Pythagorean
triple.
Since the measure of the longest side is 15, 15 must be c.
Let a and b be 9 and 12.
Pythagorean Theorem
Simplify.
Add.
Answer: These segments form the sides of a right
triangle since they satisfy the Pythagorean Theorem. The
measures are whole numbers and form a Pythagorean
triple.
Determine whether 21, 42, and 54 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Pythagorean Theorem
Simplify.
Add.
Answer: Since
, segments with these
measures cannot form a right triangle. Therefore,
they do not form a Pythagorean triple.
Determine whether
4, and 8 are the sides of a
right triangle. Then state whether they form a
Pythagorean triple.
Pythagorean Theorem
Simplify.
Add.
Answer: Since 64 = 64, segments with these measures
form a right triangle. However,
is not a whole number.
Therefore, they do not form a Pythagorean triple.
Determine whether each set of measures are the sides
of a right triangle. Then state whether they form a
Pythagorean triple.
a. 6, 8, 10
Answer: The segments form the sides of a right triangle
and the measures form a Pythagorean triple.
b. 5, 8, 9
Answer: The segments do not form the sides of a right
triangle, and the measures do not form a Pythagorean triple.
c.
Answer: The segments form the sides of a right triangle,
but the measures do not form a Pythagorean triple.
Example 1 Find the Measure of the Hypotenuse
Example 2 Find the Measure of the Legs
Example 3 30°–60°–90° Triangles
Example 4 Special Triangles in a Coordinate Plan
WALLPAPER TILING The wallpaper in the figure can
be divided into four equal square quadrants so that
each square contains 8 triangles. What is the area of
one of the squares if the hypotenuse of each
40°-45°-90° triangle measures
millimeters?
The length of the hypotenuse of one 40°-45°-90°
triangle is
millimeters. The length of the hypotenuse
is
times as long as a leg. So, the length of each leg is
7 millimeters.
The area of one of these triangles is
or 24.5 millimeters.
Answer: Since there are 8 of these triangles in one
square quadrant, the area of one of these
squares is 8(24.5) or 196 mm2.
WALLPAPER TILING If each 40°-45°-90° triangle in
the figure has a hypotenuse of
millimeters, what
is the perimeter of the entire square?
Answer: 80 mm
Find a.
The length of the hypotenuse of a 40°-45°-90° triangle
is
times as long as a leg of the triangle.
Divide each side by
Rationalize the denominator.
Multiply.
Divide.
Answer:
Find b.
Answer:
Find QR.
is the longer leg,
hypotenuse.
is the shorter leg, and
is the
Multiply each side by 2.
Answer:
Find BC.
Answer: BC = 8 in.
COORDINATE GEOMETRY
is a 30°-60°-90°
triangle with right angle X and
as the longer leg.
Graph points X(-2, 7) and Y(-7, 7), and locate point W
in Quadrant III.
Graph X and Y.
lies on a horizontal gridline of the
coordinate plane. Since
will be perpendicular to
it lies on a vertical gridline. Find the length of
is the shorter leg.
is the longer
leg. So,
Use XY to find
WX.
Point W has the same x-coordinate as X. W is located
units below X.
Answer: The coordinates of W are
or about
COORDINATE GEOMETRY
is at 30°-60°-90°
triangle with right angle R and
as the longer leg.
Graph points T(3, 3) and R(3, 6) and locate point S in
Quadrant III.
Answer: The coordinates of S are
or about
Example 1 Find Sine, Cosine, and Tangent Ratios
Example 2 Evaluate Expressions
Example 3 Use Trigonometric Ratios to Find a Length
Example 4 Use Trigonometric Ratios to Find an Angle
Measure
Find sin L, cos L, tan L, sin N, cos N, and tan N.
Express each ratio as a fraction and as a decimal.
Answer:
Find sin A, cos A, tan A, sin B, cos B, and tan B.
Express each ratio as a fraction and as a decimal.
Answer:
Use a calculator to find tan
thousandth.
KEYSTROKES: TAN
Answer:
56
to the nearest ten
ENTER
1.482560969
Use a calculator to find tan
thousandth.
KEYSTROKES: COS
Answer:
90
to the nearest ten
ENTER
0
a. Use a calculator to find sin 48° to the nearest ten
thousandth.
Answer:
b. Use a calculator to find cos 85° to the nearest ten
thousandth.
Answer:
EXERCISING A fitness trainer sets the incline on a
treadmill to
The walking surface is 5 feet long.
Approximately how many inches did the trainer raise
the end of the treadmill from the floor?
Let y be the height of the treadmill from the floor in inches.
The length of the treadmill is 5 feet, or 60 inches.
Multiply each side by 60.
Use a calculator to find y.
KEYSTROKES: 60
SIN
7
ENTER
7.312160604
Answer: The treadmill is about 7.3 inches high.
CONSTRUCTION The bottom of a handicap ramp is
15 feet from the entrance of a building. If the angle of
the ramp is about
how high does the ramp rise
off the ground to the nearest inch?
Answer: about 15 in.
COORDINATE GEOMETRY Find mX in right XYZ for
X(–2, 8), Y(–6, 4), and Z(–3, 1).
Explore You know the coordinates of the vertices of a
right triangle and that
is the right angle. You
need to find the measures of one of the angles.
Plan
Use the Distance Formula to find the measure
of each side. Then use one of the trigonometric
ratios to write an equation. Use the inverse to
find
Solve
or
or
or
Use the cosine ratio.
Simplify.
Solve for x.
Use a calculator to find
KEYSTROKES:
2ND
4
ENTER
Examine Use the sine ratio to check the answer.
Simplify.
5
)
KEYSTROKES:
2ND
ENTER
Answer: The measure of
is about 36.9.
3
5
)
COORDINATE GEOMETRY
Answer: about 56.3
Example 1 Angle of Elevation
Example 2 Angle of Depression
Example 3 Indirect Measurement
CIRCUS ACTS At the circus, a person in the audience
watches the high-wire routine. A 5-foot-6-inch tall
acrobat is standing on a platform that is 25 feet off the
ground. How far is the audience member from the
base of the platform, if the angle of elevation from the
audience member’s line of sight to the top of the
acrobat is
Make a drawing.
Since QR is 25 feet and RS is 5 feet 6 inches or 5.5 feet,
QS is 30.5 feet. Let x represent PQ.
Multiply both sides by x.
Divide both sides by tan
Simplify.
Answer: The audience member is about 60 feet
from the base of the platform.
DIVING At a diving competition, a 6-foot-tall diver
stands atop the 32-foot platform. The front edge of the
platform projects 5 feet beyond the ends of the pool.
The pool itself is 50 feet in length. A camera is set up
at the opposite end of the pool even with the pool’s
edge. If the camera is angled so that its line of sight
extends to the top of the diver’s head, what is the
camera’s angle of elevation to the nearest degree?
Answer: about
SHORT-RESPONSE TEST ITEM
A wheelchair ramp is 3 meters long and inclines at
Find the height of the ramp to the nearest tenth
centimeter.
Read the Test Item
The angle of depression between the ramp and the
horizontal is
Use trigonometry to find the height of
the ramp.
Solve the Test Item
Method 1
The ground and the horizontal level with the platform
to which the ramp extends are parallel. Therefore,
since they are alternate interior
angles.
Y
W
Mulitply each side by 3.
Simplify.
Answer: The height of the ramp is about 0.314 meters,
Method 2
The horizontal line from the top of the platform to which
the wheelchair ramp extends and the segment from the
ground to the platform are perpendicular. So,
and
are complementary angles. Therefore,
Y
W
Multiply each side by 3.
Simplify.
Answer: The height of the ramp is about 0.314 meters,
SHORT-RESPONSE TEST ITEM
A roller coaster car is at one of its highest points. It
drops at a
angle for 320 feet. How high was the
roller coaster car to the nearest foot before it began
its fall?
Answer: The roller coaster car was about 285 feet above
the ground.
Vernon is on the top deck of a cruise ship and
observes two dolphins following each other directly
away from the ship in a straight line. Vernon’s position
is 154 meters above sea level, and the angles of
depression to the two dolphins are
Find the distance between the two dolphins to the
nearest meter.
are right triangles. The distance between
the dolphins is JK or
Use the right triangles to find
these two lengths.
Because
are horizontal lines, they are parallel.
Thus,
and
because they
are alternate interior angles. This means that
Multiply each side by JL.
Divide each side by tan
Use a calculator.
Multiply each side by KL.
Divide each side by tan
Use a calculator.
Answer: The distance between the dolphins is
, or about 8 meters.
Madison looks out her second-floor window, which is
15 feet above the ground. She observes two parked
cars. One car is parked along the curb directly in front
of her window, and the other car is parked directly
across the street from the first car. The angles of
depression of Madison’s line of sight to the cars are
Find the distance between the two cars.
Answer: about 24 feet
Example 1 Use the Law of Sines
Example 2 Solve Triangles
Example 3 Indirect Measurement
Find p. Round to the nearest tenth.
Law of Sines
Cross products
Divide each side by tan
Use a calculator.
Answer:
to the nearest degree in
,
Law of Sines
Cross products
Divide each side by 7.
Solve for L.
Use a calculator.
Answer:
a. Find c.
Answer:
b. Find mT to the nearest degree in RST if r = 12,
t = 7, and mT = 76.
Answer:
. Round
angle measures to the nearest degree and side
measures to the nearest tenth.
We know the measures of two angles of the triangle. Use
the Angle Sum Theorem to find
Angle Sum Theorem
Add.
Subtract 120 from each side.
Since we know
and f, use proportions involving
To find d:
Law of Sines
Substitute.
Cross products
Divide each side by sin 8°.
Use a calculator.
To find e:
Law of Sines
Substitute.
Cross products
Divide each side by sin 8°.
Use a calculator.
Answer:
Round angle
measures to the nearest degree and side measures
to the nearest tenth.
We know the measure of two sides and an angle opposite
one of the sides.
Law of Sines
Cross products
Divide each side by 16.
Solve for L.
Use a calculator.
Angle Sum Theorem
Substitute.
Add.
Subtract 116 from each side.
Law of Sines
Cross products
Divide each side by sin
Use a calculator.
Answer:
a. Solve
Round
angle measures to the nearest degree and side
measures to the nearest tenth.
Answer:
b.
Round angle
measures to the nearest degree and side measures to
the nearest tenth.
Answer:
A 46-foot telephone pole tilted at an angle of from
the vertical casts a shadow on the ground. Find the
length of the shadow to the nearest foot when the
angle of elevation to the sun is
Draw a diagram Draw
Then find the
Since you know the measures of two angles of the
triangle,
and the length of a side
opposite one of the angles
you
can use the Law of Sines to find the length of the shadow.
Law of Sines
Cross products
Divide each side by sin
Use a calculator.
Answer: The length of the shadow is about
75.9 feet.
A 5-foot fishing pole is anchored to the edge of a
dock. If the distance from the foot of the pole to the
point where the fishing line meets the water is 45 feet,
about how much fishing line that is cast out is above
the surface of the water?
Answer: About 42 feet of the fishing line that is cast out
is above the surface of the water.
Example 1 Two Sides and the Included Angle
Example 2 Three Sides
Example 3 Select a Strategy
Example 4 Use Law of Cosines to Solve Problems
Use the Law of Cosines since the measures of two sides
and the included angle are known.
Law of Cosines
Simplify.
Take the square root
of each side.
Use a calculator.
Answer:
Answer:
Law of Cosines
Simplify.
Subtract 754 from each side.
Divide each side by –270.
Solve for L.
Use a calculator.
Answer:
Answer:
Determine whether the Law of Sines or the Law of
Cosines should be used first to solve
Then
solve
Round angle measures to the nearest
degree and side measures to the nearest tenth.
Since we know the measures of two sides and the
included angle, use the Law of Cosines.
Law of Cosines
Take the square root
of each side.
Use a calculator.
Next, we can find
If we decide to find
we can use either the Law of Sines or the Law of Cosines
to find this value. In this case, we will use the Law of
Sines.
Law of Sines
Cross products
Divide each side by 46.9.
Take the inverse of each
side.
Use a calculator.
Use the Angle Sum Theorem to find
Angle Sum Theorem
Subtract 168 from each
side.
Answer:
Determine whether the Law of Sines or the Law of
Cosines should be used first to solve
Then
solve
Round angle measures to the nearest
degree and side measures to the nearest tenth.
Answer:
AIRCRAFT From the diagram of the plane shown,
determine the approximate exterior perimeter of each
wing. Round to the nearest tenth meter.
Since
is an isosceles triangle,
Use the Law of Sines to find KJ.
Law of Sines
Cross products
Divide each side by sin
Simplify.
.
Use the Law of Sines to find
.
Law of Sines
Cross products
Divide each side by 9.
Solve for H.
Use a calculator.
Use the Angle Sum Theorem to find
Angle Sum Theorem
Subtract 95 from each
side.
Use the Law of Sines to find HK.
Law of Sines
Cross products
Divide each side by sin
Use a calculator.
The perimeter of the wing is equal to
Answer: The perimeter is about
about 67.1 meters.
or
The rear side window of a station wagon has the
shape shown in the figure. Find the perimeter of the
window if the length of DB is 31 inches.
Answer: about 93.5 in.
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