confidential1 geometry similarity in right triangles
TRANSCRIPT
CONFIDENTIAL 1
Geometry
Similarity in Right Triangles
CONFIDENTIAL 2
Warm up
Find the x- intercept and y-intercept for each equation.
1). 3y + 4 = 6x
2). x + 4 = 2y
3). 3y – 15 = 15x
1) x- intercept= 2/3 y-intercept =-4/32) x- intercept= -4 y-intercept =23) x- intercept= 5 y-intercept =-1
CONFIDENTIAL 3
In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles.
Theorem 1.1
The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.
∆ABC ~ ∆ACD ~ ∆CBD
DB
AC
Similarity in Right Triangles
CONFIDENTIAL 4
Theorem 1.2
DB
AC
Given: ∆ABC is a right triangle with altitude CD.Prove: ∆ABC ~ ∆ACD ~ ∆CBD
Proof: The right angles in ∆ABC, ∆ACD, and ∆CBDAre all congruent. By the Reflexive Property of Congruence, A ≅ A. Therefore ∆ABC ~ ∆ACD by the AA Similarity Theorem. Similarly, B ≅ B, so ∆ABC ~ ∆CBD. By the Transitive Property of Similarity, ∆ABC~∆ACD~∆CBD.
CONFIDENTIAL 5
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles of the triangles in corresponding positions.
SS
SR
R
T PP
P
T S
R
Identifying Similar Right Triangles
CONFIDENTIAL 6
Consider the proportion a = x. In this case, the means
of the proportion are the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √ab, or x2=ab.
x b
CONFIDENTIAL 7
Finding Geometric Means
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
A ) 4 and 9
Let x be the geometric mean.
x2 = (4)(9) = 36 x=6
Def. of geometric mean
Find the positive square root
B ) 6 and 15
Let x be the geometric mean.
x2=(6)(15) = 90 x= 90 =3 10 Def. of geometric mean
Find the positive square root
CONFIDENTIAL 8
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
1a) 2 and 8
1b) 10 and 30
1c) 8 and 9
Now you try!
1a) 4 1b) 10√31c) 6√2
CONFIDENTIAL 9
Theorem 1.1: to write proportions comparing the side lengths of the triangles formed by the altitude to
the hypotenuse of a right triangle. All the relationships in red involve geometric means.
c
b =
b
h=
a
hc
a =
b
h =
a
x b
a =
y
h =
h
x
c
a
h
x
b
y
hh
y
x
b
a C
C
C
D
D D
B
B
AA
CONFIDENTIAL 10
Corollaries
COROLLARY EXAMPLE DIAGRAM
a
b
x
y
h
c
1.2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of two segments of the hypotenuse.
1.3 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
h2 = xy
a2 = xcb2 = yc
CONFIDENTIAL 11
Finding Side Lengths in right Triangles
Find x, y, and z.
x2 = (2) (10) = 20 x = 20 = 2 5 y2 = (12)(10) = 120
y = 120 = 2 30
z2 = (12)(2) = 24
z = 24 = 2 6
2
10
z
y
x
X is the geometric mean of 2 and 10.
Find the positive square root.
Y is the geometric mean of 12 and 10.
Find the positive square root.
Z is the geometric mean of 12 and 2.
Find the positive square root.
CONFIDENTIAL 12
2) Find u, v, and w.
9
3
vw
u
Now you try!
2) u = 27, v = 3√10, w = 9√10
CONFIDENTIAL 13
Measurement Application
To estimate the height of Big Tex at the State Fair of Texas, Michael steps away
from the statue until his line of sight to the top of the status and his line of sight to the bottom of the statue form a 90˚ angle. His eyes are 5 ft above the ground, and he is standing 15 ft 3 in. from Big Tex. How tall
is Big Tex to the nearest foot?
Let x be the height of Big Tex above eye level.
15 ft 3 in. = 15.25 ft
(15.25) = 5x
X = 46.5125 = 47
Big Tex is about 47 + 5, or 52 ft tall.
Convert 3 in. to 0.25 ft. 15.25 is the geometric mean of 5 and x.
Solve for x and round.
5 ft
15 ft 3 in.
CONFIDENTIAL 14
3) A surveyor positions himself so that his line of sight to the top of a cliff and his line of sight to the bottom from a right angle as shown. What is the
height of the cliff to the nearest foot?
28 ft
Now you try!
3) 148 ft
5.5 ft
CONFIDENTIAL 15
Now some problems for you to practice !
CONFIDENTIAL 16
Assessment
Write a similarity statement comparing the three
triangles in each diagram.
SQR
PB
C
E D
1) 2)
CONFIDENTIAL 17
Find the geometric mean of each pair of numbers. If
necessary, give the answer in simplest radical form.
3). 2 and 50
4). 9 and 12
5). ½ and 8
3) 10 4) 6√35) 2
CONFIDENTIAL 18
Find x, y, and z.
zy
46
x
20
y
x
z 10
7).6).
6) x = 2√15, y = 2√6, z = 2√107) x = 5, y = 10√5, z = 5√5
CONFIDENTIAL 19
8) Measurement To estimate the length of the US Constitution in Boston harbor, a student located points
T and U as shown. What is RS to the nearest tenth?
4 m
60 m
S
T
U
R
8) 16√15
CONFIDENTIAL 20
In a right triangle, an altitude drawn from the vertex of the right angle to the hypotenuse forms two right triangles.
Theorem 1.1
The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.
∆ABC ~ ∆ACD ~ ∆CBD
DB
AC
Similarity in Right Triangles
Let’s review
CONFIDENTIAL 21
Theorem 1.2
DB
AC
Given: ∆ABC is a right triangle with altitude CD.Prove: ∆ABC ~ ∆ACD ~ ∆CBD
Proof: The right angles in ∆ABC, ∆ACD, and ∆CBDAre all congruent. By the Reflexive Property of Congruence, A ≅ A. Therefore ∆ABC ~ ∆ACD by the AA Similarity Theorem. Similarly, B ≅ B, so ∆ABC ~ ∆CBD. By the Transitive Property of Similarity, ∆ABC~∆ACD~∆CBD.
CONFIDENTIAL 22
Write a similarity statement comparing the three triangles.
Sketch the three right triangles with the angles of the triangles in corresponding positions.
SS
SR
R
T PP
P
T S
R
Identifying Similar Right Triangles
CONFIDENTIAL 23
Finding Geometric Means
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
A ) 4 and 9
Let x be the geometric mean.
x2 = (4)(9) = 36 x=6
Def. of geometric mean
Find the positive square root
B ) 6 and 15
Let x be the geometric mean.
x2=(6)(15) = 90 x= 90 =3 10 Def. of geometric mean
Find the positive square root
CONFIDENTIAL 24
Theorem 1.1: to write proportions comparing the side lengths of the triangles formed by the altitude to
the hypotenuse of a right triangle. All the relationships in red involve geometric means.
c
b =
b
h=
a
hc
a =
b
h =
a
x b
a =
y
h =
h
x
c
a
h
x
b
y
hh
y
x
b
a C
C
C
D
D D
B
B
AA
CONFIDENTIAL 25
Corollaries
COROLLARY EXAMPLE DIAGRAM
a
b
x
y
h
c
1.2 The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of two segments of the hypotenuse.
1.3 The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
h2 = xy
a2 = xcb2 = yc
CONFIDENTIAL 26
Finding Side Lengths in right Triangles
Find x, y, and z.
x2 = (2) (10) = 20 x = 20 = 2 5 y2 = (12)(10) = 120
y = 120 = 2 30
z2 = (12)(2) = 24
z = 24 = 2 6
2
10
z
y
x
X is the geometric mean of 2 and 10.
Find the positive square root.
Y is the geometric mean of 12 and 10.
Find the positive square root.
Z is the geometric mean of 12 and 2.
Find the positive square root.
CONFIDENTIAL 27
Measurement Application
To estimate the height of Big Tex at the State Fair of Texas, Michael steps away
from the statue until his line of sight to the top of the status and his line of sight to the bottom of the statue form a 90˚ angle. His eyes are 5 ft above the ground, and he is standing 15 ft 3 in. from Big Tex. How tall
is Big Tex to the nearest foot?
Let x be the height of Big Tex above eye level.
15 ft 3 in. = 15.25 ft
(15.25) = 5x
X = 46.5125 = 47
Big Tex is about 47 + 5, or 52 ft tall.
Convert 3 in. to 0.25 ft. 15.25 is the geometric mean of 5 and x.
Solve for x and round.
5 ft
15 ft 3 in.
CONFIDENTIAL 28
You did a great job today!You did a great job today!