Algebra 1 Interactive Chalkboard
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Lesson 11-1Simplifying Radical Expressions
Lesson 11-2Operations with Radical Expressions
Lesson 11-3Radical Equations
Lesson 11-4The Pythagorean Theorem
Lesson 11-5The Distance Formula
Lesson 11-6Similar Triangles
Lesson 11-7Trigonometric Ratios
Example 1 Simplify Square Roots
Example 2 Multiply Square Roots
Example 3 Simplify a Square Root with Variables
Example 4 Rationalizing the Denominator
Example 5 Use Conjugates to Rationalize a
Denominator
Simplify
.
Prime factorization of 52
Product Property of Square Roots
Answer:
Simplify.
Simplify
.
Prime factorization of 72
Product Property of Square Roots
Simplify.
Answer:
Simplify.
Simplify.
a.
Answer:
b.
Answer:
Find
Product Property
Product Property
Answer:
Simplify.
Find
Answer:
Simplify
Prime factorization
Product Property
Simplify.
Answer:
The absolute value
of
ensures a
nonnegative result.
Simplify
Answer:
Simplify
.
Multiply by
.
Product Property of Square Roots
Answer:
Simplify.
Simplify
.
Prime factorization
Multiply by
Answer:
Product Property of Square Roots
Simplify
.
Multiply by
Product Property of Square Roots
Prime factorization
Answer:
Divide the numerator and
denominator by 2.
Simplify.
a.
Answer:
b.
Answer:
c.
Answer:
Simplify
Answer:
Simplify.
Simplify
Answer:
Example 1 Expressions with Like Radicands
Example 2 Expressions with Unlike Radicands
Example 3 Multiply Radical Expressions
Simplify
.
Distributive Property
Answer:
Simplify.
Simplify
.
Commutative Property
Distributive Property
Answer:
Simplify.
Simplify
a.
Answer:
b.
Answer:
Simplify
Answer: The simplified form is
Simplify
Answer:
Find the area of a rectangle with a width of
and a length of
To find the area of the rectangle multiply the measures of
the length and width.
First terms
Outer terms
Inner terms
Last terms
Multiply.
Prime factorization
Simplify.
Combine like
terms.
Answer: The area of the rectangle is
square units.
Find the area of a rectangle with a width of
and a length of
Answer: 26
+ 106
units2
Example 1 Radical Equation with a Variable
Example 2 Radical Equation with an Expression
Example 3 Variable on Each Side
Free-Fall Height An object is dropped from an
unknown height and reaches the ground in 5 seconds.
From what height is it dropped?
Use the equation
and replace t with 5 seconds.
Original equation
Replace t with 5.
Multiply each side by 4.
Square each side.
Simplify.
Check
Original equation
and
Answer: The object is dropped from 400 feet.
Free-Fall Height An object is dropped from an
unknown height and reaches the ground in 7 seconds.
Use the equation
it is dropped.
Answer: 784 ft
to find from what height
Solve
Original equation
Subtract 8 from each side.
Square each side.
Add 3 to each side.
Answer: The solution is 52.
Solve
Answer: 60
Solve
Original equation
Square each side.
Simplify.
Subtract 2 and add y to
each side.
Factor.
or
Zero Product Property
Solve.
Check
Answer: Since –2 does not satisfy the original equation,
1 is the only solution.
Solve
Answer: 3
Example 1 Find the Length of the Hypotenuse
Example 2 Find the Length of a Side
Example 3 Pythagorean Triples
Example 4 Check for Right Triangles
Find the length of the hypotenuse of a right triangle if
and
Pythagorean Theorem
and
Simplify.
Take the square root of each side.
Use the positive value.
Answer: The length of the hypotenuse is 30 units.
Find the length of the hypotenuse of a right triangle if
and
Answer: 65 units
Find the length of the missing side.
In the triangle,
and
units.
Pythagorean Theorem
and
Evaluate squares.
Subtract 81 from each side.
Use a calculator to evaluate
Use the positive value.
Answer: To the nearest hundredth, the length of
the leg is 13.23 units.
.
Find the length of the missing side.
Answer: about 16.25 units
Multiple-Choice Test Item
What is the area of triangle XYZ?
A
B
C
D
94 units2
2
128 units
294 units2
588 units2
Read the Test Item
The area of the triangle is
In a right triangle,
the legs form the base and height of the triangle. Use
the measures of the hypotenuse and the base to
find the height of the triangle.
Solve the Test Item
Step 1 Check to see if the measurements of this triangle
are a multiple of a common Pythagorean triple.
The hypotenuse is
units and the leg is
units.
This triangle is a multiple of a (3, 4, 5) triangle.
The height of the triangle is 21 units.
Step 2 Find the area of the triangle.
Area of a triangle
and
Simplify.
Answer: The area of the triangle is 294 square units.
Choice C is correct.
Multiple-Choice Test Item
What is the area of triangle RST?
A
B
C
D
764 units2
2
480 units
420 units2
384 units2
Answer: D
Determine whether the side measures of 7, 12, 15
form a right triangle.
Since the measure of the longest side is 15, let
, and
Then determine whether
Pythagorean Theorem
and
Multiply.
Add.
Answer: Since
right triangle.
, the triangle is not a
Determine whether the side measures of 27, 36, 45
form a right triangle.
Since the measure of the longest side is 45, let
and
Then determine whether
Pythagorean Theorem
and
Multiply.
Add.
Answer: Since
right triangle.
the triangle is a
Determine whether the following side measures form
right triangles.
a. 33, 44, 55
Answer: right triangle
b. 12, 22, 40
Answer: not a right triangle
Example 1 Distance Between Two Points
Example 2 Use the Distance Formula
Example 3 Find a Missing Coordinate
Find the distance between the points at
(1, 2) and (–3, 0).
Distance Formula
and
Simplify.
Evaluate squares
and simplify.
Answer:
or about 4.47 units
Find the distance between the points at
(5, 4) and (0, –2).
Answer:
Biathlon Julianne is sighting her rifle for an
upcoming biathlon competition. Her first shot is 2
inches to the right and 7 inches below the bull’s-eye.
What is the distance between the bull’s-eye and
where her first shot hit the target?
Draw a model of the situation on a coordinate grid.
If the bull’s-eye is at (0, 0), then the location of the first
shot is (2, –7). Use the Distance Formula.
Distance Formula
and
Simplify.
or about 7.28 inches
Answer: The distance is
or about 7.28 inches.
Horseshoes Marcy is pitching a horseshoe in her
local park. Her first pitch is 9 inches to the left and 3
inches below the pin. What is the distance between
the horseshoe and the pin?
Answer:
Find the value of a if the distance between the points
at (2, –1) and (a, –4) is 5 units.
Distance Formula
Let
and
Simplify.
Evaluate squares.
Simplify.
.
Square each side.
Subtract 25 from each side.
Factor.
or
Zero Product Property
Solve.
Answer: The value of a is –2 or 6.
Find the value of a if the distance between the points
at (2, 3) and (a, 2) is
Answer: –4 or 8
Example 1 Determine Whether Two Triangles
Are Similar
Example 2 Find Missing Measures
Example 3 Use Similar Triangles to Solve a Problem
Determine whether the pair
of triangles is similar. Justify
your answer.
The ratio of sides XY to AB is
The ratio of sides YZ to BC is
The ratio of sides XZ to AC is
Answer: Since the measures of the corresponding
sides are proportional, triangle XYZ is similar to
triangle ABC.
Determine whether the pair of triangles is similar.
Justify your answer.
Answer: Since the corresponding angles have equal
measures, the triangles are similar.
Find the missing measures if the pair of triangles
is similar.
Since the corresponding
angles have equal measures,
The lengths
of the corresponding sides
are proportional.
Corresponding sides of similar
triangles are proportional.
and
Find the cross products.
Divide each side by 18.
Corresponding sides of similar
triangles are proportional.
and
Find the cross products.
Divide each side by 18.
Answer: The missing measures are 27 and 12.
Find the missing measures if the pair
of triangles is similar.
Corresponding
sides of similar
triangles are
proportional.
and
Find the cross products.
Divide each side by 4.
Answer: The missing measure is 7.5.
Find the missing measures if each pair of triangles
is similar.
a.
Answer: The missing measures are 18 and 42.
Find the missing measures if each pair of triangles
is similar.
b.
a
Answer: The missing measure is 5.25 mm.
Shadows Richard is standing next to the General
Sherman Giant Sequoia tree in Sequoia National Park.
The shadow of the tree is 22.5 meters, and Richard’s
shadow is 53.6 centimeters. If Richard’s height is 2
meters, how tall is the tree?
Since the length of the shadow of the tree and Richard’s
height are given in meters, convert the length of Richard’s
shadow to meters.
Simplify.
Let
the height of the tree.
Richard’s shadow
Richard’s height
Tree’s shadow
Tree’s height
Cross products
Answer: The tree is about 84 meters tall.
Tourism Trudie is standing next to the Eiffel Tower in
France. The height of the Eiffel Tower is 317 meters
and casts a shadow of 155 meters. If Trudie’s height
is 2 meters, how long is her shadow?
Answer: The length of Trudie’s shadow is about
0.98 meter.
Example 1 Sine, Cosine, and Tangent
Example 2 Find the Sine of an Angle
Example 3 Find the Measure of an Angle
Example 4 Solve a Triangle
Example 5 Angle of Elevation
Find the sine, cosine, and tangent of each acute angle
of
Round to the nearest ten-thousandth.
Write each ratio and substitute the measures. Use a
calculator to find each value.
Answers:
Answer:
1.0659
Answers:
Answer:
Find the sine, cosine, and tangent of each acute angle
of
Round to the nearest ten-thousandth.
Answer:
Find cos 65° to the nearest ten thousandth.
Keystrokes COS 65
ENTER .4226182617
Answer: Rounded to the nearest ten thousandth,
Find tan 32° to the nearest ten thousandth.
Answer: 0.6249
Find the measure of
to the nearest degree.
Since the lengths of the adjacent leg and
the hypotenuse are known, use the
cosine ratio.
and
Now use [COS–1] on a calculator to find the measure of
the angle whose cosine ratio is
Keystrokes 2nd [COS–1] 12
20
ENTER
53.13010235
Answer: To the nearest degree, the measure of
is 53°.
Find the measure of
Answer: 29°
to the nearest degree.
Find all of the missing measures in
You need to find the measures of
and
Step 1 Find the measure of
The
sum of the measures of the
angles in a triangle is 180.
The measure of
is 28°.
Step 2 Find the value of y, which is the measure of the
hypotenuse. Since you know the measure of the
side opposite
, use the sine ratio.
Definition of sine
Evaluate sin 62°.
Multiply each side by y
and divide each side by
0.8829.
is about 17.0 centimeters long.
Step 3 Find the value of x, which is the measure of the side
adjacent
Use the tangent ratio.
tan 28º
Definition of tangent
0.5317
Evaluate tan 28°.
Find the cross products.
is about 8.0 centimeters long.
Answer: So, the missing measures are 28,
8 cm, and 17 cm.
Find all of the missing measures in
Answer: The missing measures are 47, 11 m,
and 16 m.
Indirect Measurement In the diagram, Barone is flying
his model airplane 400 feet above him. An angle of
depression is formed by a horizontal line of sight and
a line of sight below it. Find the angles of depression
at points A and B to the nearest degree.
Explore In the diagram two right
triangles are formed. You
know the height of the
airplane and the horizontal
distance it has traveled.
Plan
Let A represent the first
angle of depression. Let
B represent the second
angle of depression.
Solve
Write two equations involving the tangent ratio.
and
Answer: The angle of depression at point A is 45° and
the angle of depression at point B is 37°.
Examine Examine the solution by finding the horizontal
distance the airplane has flown at points A and B.
The solution checks.
Indirect Measurement In the diagram, Kylie is flying a
kite 350 feet above her. An angle of depression is
formed by a horizontal line of sight and a line of sight
below it. Find the angle of depression at points X and
Y to the nearest degree.
Answer: The angle of
depression at
point X is 38°
and the angle
of depression
at Y is 32°.
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