michael reyes mted 301 section 1-2. subject: geometry grade level:9-10 lesson:
DESCRIPTION
Michael Reyes MTED 301 Section 1-2. Subject: Geometry Grade Level:9-10 Lesson: The Distance Formula Objective: California Mathematics Content Standard Geometry 17.0: - PowerPoint PPT PresentationTRANSCRIPT
Michael ReyesMTED 301 Section 1-2.Subject: GeometryGrade Level:9-10Lesson:The Distance FormulaObjective:California Mathematics Content StandardGeometry 17.0:Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.California Common Core StandardGeometry Congruence 9.0:Prove theorems about lines and angles. Materials: Larson, R., Boswell, L., Kanold, T., Stiff, L. McDougall Littell.(2001). Algebra I, pp. 745-751.
Warm-Up
1. Plot and label A(2,1) and B(6,5) and C(6,1) on graph paper. Connect the points to form a right triangle with AB as the hypotenuse.
2. Find the lengths of the legs of triangle ABC. This means find the lengths of BC and CA.
3. Use the Pythagorean theorem to find the length of the hypotenuse AB.
Warm-Up cont.
4. Solve the expression. Round your final answer to the nearest hundredths.
)2530()35(
Warm-Up Solution to #1
A(2,1) C(6,1
)
B(6,4)
AB
Warm-Up Solution to #2
Find the lengths of the legs of ABC.BC 12 yy
15 BC4BC
CA 12 xx
24 CA
2CA
Warm-Up Solution to #3
Use the Pythagorean theorem to find the length of the hypotenuse AB. 222
CABCAB
22234 AB
9162
AB
252AB
25AB
5AB
Warm-Up Solution to #4
4.
2. 65
The Distance Formula
The steps used in the warm up can be used to develop a general formula for the distance between two points.
),(
),(
22
11
yxB
yxA
The Distance Formula
What are the coordinates of C?
),(
),(
22
11
yxB
yxA
),( 11 yxA
),( 22 yxB
C
),( 12 yxC
The Distance Formula
What are the coordinates of C?
),(
),(
22
11
yxB
yxA
),( 11 yxA
),( 22 yxB
),( 12 yxC
),( 12 yxC
The Distance Formula
What are the lengths of the triangle’s sides?
),( 11 yxA
),( 22 yxB
12 xxAC
12 yyBC ),( 12 yxC
12 xxAC
12 yyBC
The Distance Formula
What is the length of the triangle’s hypotenuse?
),( 11 yxA
),( 22 yxB
),( 12 yxC
12 xxAC
12 yyBC 222
BCACAB
2122
12
2yyxxAB
2122
12 yyxxAB The length of the hypotenuse is equal to the distance between points A and B.
2122
12 yyxxd
AB
Vocabulary Check
The _____________can be obtained by creating a triangle and using the ________________to find the length of the hypotenuse. The hypotenuse of the triangle will be the ___________ between the two points.
Distance Formula
Pythagorean Theorem
Distance
Example 1Find the distance between (1,4)
and (-2,3).Solution:
2122
12 )( yyxxd
22 43)12( d
10d16.3d
Write the distance formulaSubstitute
Simplify
Use a calculator
Example 2Find the distance between
.1,24
1,2
1and
Example 2
Solution: 212
212 )( yyxxd Write the distance
formula22
4
11
2
12
d Substitute
22
4
3
2
3
d
16
45d
67.1d
Simplify
Simplify
Use a calculator
Example 3 Checking A Right Triangle
Decide whether the points (3,2),(2,0), and (-1,4) are vertices of a right triangle.Begin by graphing the triangle with the given vertices.
Example 3 Checking A Right Triangle
(-1,4)
(2,0)
(3,2)d2
d3 d1
Does this look like a right triangle? We can apply the distance formula to check if it is truly a right triangle.
Example 3 Checking A Right Triangle
Solution:Use the distance formula to find the lengths of the three sides.
22 )02()23(1 d
22 )40())1(2(3 d
22 )42())1(3(2 d 416
169
41 5
20
25
Example 3 Checking A Right Triangle
Next we find the sum of the squares of the lengths of the two shorter sides.
(-1,4)
(2,0)
(3,2)d2
d3 d1
2222 20521 dd
20525
Example 3 Checking A Right Triangle
The sum of the squares of the lengths of the shorter sides is 25.
This is equal to the square of the length of the longest side,
(-1,4)
(2,0)
(3,2)d2
d3 d1
2222 20521 dd
20525
225Thus, the given points are vertices of a right triangles.
Example 4 Application of the Distance Formula
How can you use the distance formula to solve problems like the following one:The point (1,2) lies on a circle. What is the length of the radius of this circle if the center is located at (4,6)?
Example 4 Application of the Distance Formula
How can you use the distance formula to solve problems like the following one:The point (1,2) lies on a circle. What is the length of the radius of this circle if the center is located at (4,6)?
(1,2)
(4,6)
Circles Review
Example 4 Application of the Distance Formula
The point (1,2) lies on a circle. What is the length of the radius of this circle if the center is located at (4,6)?The length is equal to the distance between the center point and any point located on the edge of the circle.
(1,2)
(4,6)
Example 4 Application of the Distance Formula
radius=distance(d)
(1,2)
(4,6)
2122
12 )( yyxxd
22 26)14( d
22 4)3( d
169d
25d5d
Solve the following individually:1. Find the distance between(-3,4)
and (5,4).2. Find the distance between the
two points:
3. The point (5,4) lies on a circle. What is the length of the radius of this circle if the center is located at (3,2)?
.3
8,3
2,
6
1,3
1
Solutions to individual problem #1
Find the distance between(-3,4) and (5,4).
Solution:
2122
12 )( yyxxd
22 44))3(5( d2)8(d
8d
Solutions to individual problem #2
2122
12 )( yyxxd 22
6
1
3
8
3
1
3
2
d
22
6
116
3
12
d
22
6
15
3
3
d
4
251d
4
25
4
4d
4
29d
2. Find the distance between the two points:
Help with Fractions
Solutions to individual problem #3
The point (5,4) lies on a circle. What is the length of the radius of this circle if the center is located at (3,2)?
(5,4)
(3,2)
Solutions to individual problem #3
(5,4)
(3,2)
Distance(d)=length of radius
2122
12 )( yyxxd
22 24)35( d
22 2)2( d
44d
8d
22d24 d
22r
Solve the following problem with a partner:1. Draw the polygon whose
vertices are A(1,1),B(5,9),C(2,8), and D(0,4).
2. Show that the polygon is a trapezoid by showing that only two of the sides are parallel.
3. Use the distance formula to show that the trapezoid is isosceles.
Trapezoid Review
Solution to trapezoid problem #1
Draw the polygon whose vertices are A(1,1),B(5,9),C(2,8), and D(0,4).
A(1,1)
B(5,9)C(2,8)
D(0,4)
Solution to trapezoid problem #2
Show that the polygon is a trapezoid by showing that only two of the sides are parallel.
A(1,1)
B(5,9)C(2,8)
D(0,4)
Slope Review
Solution to trapezoid problem #2
Show that the polygon is a trapezoid by showing that only two of the sides are parallel.
A(1,1)
B(5,9)C(2,8)
D(0,4)
Solution to trapezoid problem #2
Two side are parallel if they have the same slope.
CB has a positive slope and DA has a negative slope.
The slopes that are left to check are those of AB and CD
A(1,1)
B(5,9)C(2,8)
D(0,4)
Solution to trapezoid problem #2
Slope of AB:
Slope of CD:
A(1,1)
B(5,9)C(2,8)
D(0,4)
15
19
m
02
48
m
2m
2m
Since the slopes of AB and CD are equal, then the two sides are parallel. The polygon is a trapezoid by definition of a trapezoid.
2m
2m
Solution to trapezoid problem #3
Use the distance formula to show that the trapezoid is isosceles.We must show that CB and DA have the same distance to demonstrate that trapezoid ABCD is isosceles.
A(1,1)
B(5,9)C(2,8)
D(0,4)
Solution to trapezoid problem #3
A(1,1)
B(5,9)C(2,8)
D(0,4)
22 98)52( CB
22 1)3( CB
19CB
10CB
22 14)10( DA
22 3)1( DA
91DA
10DA10DACB
Since CB and DA are equidistant, the trapezoid ABCD is isosceles.
What we learned today:How the distance formula is
derived.How to find the distance between
to points.How to check if a triangle is a
right triangle using the distance formula
How to prove properties of shapes using the distance formula.
Fill in the blanks and turn in before you leave.
The distance formula can be obtained by creating a triangle and using the ________________to find the length of the hypotenuse. The hypotenuse of the triangle will be the ___________ between the two points.The distance between the center of a circle and any point on the circle is the ______ of the circle.
The End
Help with fractionsAdd the following:
25
9
5
4
25
9
55
54
25
9
25
20
25
29
Help with FractionsReduce the expression
25
12
25
12
5
34
5
32Back to problem
Circles ReviewWhat is the
definition of a circle?
A circle is the set of all points equidistant from a center point.
AD, BD, and CD are all equidistant and radii of the circle.
C
B
AD
Back to Lesson
Slope Review
run
riseslope
12
12
xx
yyslope
Slope Review
(7,9)
(1,1)
Find the slope of the line that passes through the points (7,9) and (1,1).
12
12
xx
yyslope
17
19
slope
6
8slope
3
4slope
The slope is positive. Note that when a line has a positive slope it rises up left to right.
Slope Review
(7,2)
(1,5)
Find the slope of the line that passes through the points (1,5) and (7,2).
12
12
xx
yyslope
71
25
slope
6
3
slope
2
1slope
The slope is negative. Note that when a line has a negative slope it falls left to right.
Back to Lesson
Trapezoid Review
A trapezoid is a quadrilateral with two sides parallel.If both legs are the same length, this is called an isosceles trapezoid, and both base angles are the same. A
B
C
D
A and C are parallel; they have the same slope.
If B and D have the same length, then the trapezoid is isosceles.
Back to Lesson