similar right triangles 1. when we use a mirror to view the top of something……
TRANSCRIPT
Similar Right Triangles1. When we use a mirror to view
the top of something…….
Similar Right Triangles1. When we use a mirror to view
the top of something…….
Eddie places a mirror 500 metersfrom a large Iron structure
(mirror)
Similar Right Triangles1. When we use a mirror to view
the top of something…….
Eddie places a mirror 500 metersfrom a large Iron structure
(mirror)
His eyes are 1.8 metersabove the ground
Similar Right Triangles1. When we use a mirror to view
the top of something…….
Eddie places a mirror 500 metersfrom a large Iron structure
(mirror)
His eyes are 1.8 metersabove the ground
He stands 2.75 meters behindthe mirror and sees the top
Similar Right Triangles1. When we use a mirror to view
the top of something…….Eddie places a mirror 500 metersfrom a large Iron structure
His eyes are 1.8 metersabove the ground
He stands 2.75 meters behindthe mirror and sees the topx
500 m 2.75 m
1.8 m
Similar Right TrianglesThe triangles are similar by Angle-Angle (AA)
Since the triangles are similar…….
x 1.8
x
500 m
1.8 m
500 2.75=
2.75 m
Similar Right TrianglesThe triangles are similar by Angle-Angle (AA)
Since the triangles are similar…….
x 1.8
x = 500 1.8 2.75
x
500 m
1.8 m
500 2.75=
.
2.75 m
Similar Right TrianglesThe triangles are similar by Angle-Angle (AA)
Since the triangles are similar…….
x 1.8
x = 500 1.8 2.75
x
500 m
1.8 m
500 2.75=
.
x = 328 m
2.75 m
Similar Right TrianglesThe triangles are similar by Angle-Angle (AA)
But since the triangles are similar we could equally say…
x 500
x = 500 1.8 2.75
x
500 m
1.8 m
1.8 2.75=
.
x = 328 m
2.75 m
Similar Right Triangles2. Now try another one………..
At a certain time of day, a lighthouse casts a 200 meter shadow
Similar Right Triangles2. Now try another one………..
At a certain time of day, a lighthouse casts a 200 meter shadow
At the same time Eddie castsA 3.1 meter shadow.
Similar Right Triangles2. Now try another one………..
At a certain time of day, a lighthouse casts a 200 meter shadow
His head is 1.9 metersabove the ground
At the same time Eddie castsA 3.1 meter shadow.
How high is theLighthouse?
Similar Right Triangles2. Now try another one………..
At a certain time of day, a lighthouse casts a 200 meter shadow
His head is 1.9 metersabove the ground
At the same time Eddie castsA 3.1 meter shadow.
How high is theLighthouse?
200 m 3.1 m
1.9 m
x
Similar Right Triangles
200 m 3.1 m
1.9 m
x 1.9
x = 200 1.9 3.1
x
200 3.1=
.
x = 123 m
The triangles are similar by Angle-Angle (AA)
Section 10-3:
• Theorem 10 - 3: “The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original and each other”
Section 10-3:
• Theorem 10 - 3: “The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original and each other”
A
B
CD
A
B C
D
D
A
B
D
B
C
ΔABC ~ ΔADB ~ ΔBDC
A
B C
D
D
A
B
D
B
C
Similar Triangles, so…….
AD BDBD CD
=
ΔABC ~ ΔADB ~ ΔADC
A
B C
D
D
A
B
D
B
C
BD is the geometric mean of AD and CD
COROLLARY 1
AD BDBD CD
=
• The altitude to the hypotenuse of ΔABC is 4 cm• If the distance AD is 2 cm, find the distance CD.
Corollary 1:The altitude to the hypotenuse is the geometric mean
of the two sections it splits the hypotenuse into
A
B
CD
• The altitude to the hypotenuse of ΔABC is 4 cm• If the distance AD is 2 cm, find the distance CD.
• . x = 4 4
Corollary 1:The altitude to the hypotenuse is the geometric mean
of the two sections it splits the hypotenuse into
A
B
CD
4 cm
2 cm
x 4 4 2
=
x cm
2.
• The altitude to the hypotenuse of ΔABC is 4 cm• If the distance AD is 2 cm, find the distance CD.
• . x = 4 4
• . x = 8 cm
Corollary 1:The altitude to the hypotenuse is the geometric mean
of the two sections it splits the hypotenuse into
A
B
CD
4 cm
2 cm
x 4 4 2
=
x cm
2.
• The altitude to the hypotenuse of ΔABC cuts AC into sections that are 4 cm long and 5 cm long
• Find the area of ΔABC
Corollary 1:The altitude to the hypotenuse is the geometric mean
of the two sections it splits the hypotenuse into
A
B
CD
A
B C
D
D
A
B
D
B
C
ΔABC ~ ΔADB ~ ΔBDC
COROLLARY 2 AC BCBC CD
ANDAC ABAB AD
D
B
=
• The altitude to the hypotenuse of ΔABC cuts AC into sections that are 3 cm long and 6 cm long
• Find the length of the legs AB and BC.
Corollary 2:Each leg of the large triangle is the geometric mean of the
hypotenuse and the adjacent segment of hypotenuse
A
B
CD
• Find the length of the legs AB and BC.
• Hypotenuse, AC = 9 cm
Corollary 2:Each leg of the large triangle is the geometric mean of the
hypotenuse and the adjacent segment of hypotenuse
A
B
CD3 cm 6 cm
x y
• Find the length of the legs AB and BC.
• Hypotenuse, AC = 9 cm
Corollary 2:Each leg of the large triangle is the geometric mean of the
hypotenuse and the adjacent segment of hypotenuse
A
B
CD3 cm 6 cm
x y
3393 x
• Find the length of the legs AB and BC.
• Hypotenuse, AC = 9 cm
Corollary 2:Each leg of the large triangle is the geometric mean of the
hypotenuse and the adjacent segment of hypotenuse
A
B
CD3 cm 6 cm
x y
3393 x 6396 y