Geo 9 3 Altitude on Hypotenuse Thm ppt comp

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Geo 9 3 Altitude on Hypotenuse Thm ppt comp Powered By Docstoc
					9.3 Altitude-on-Hypotenuse
Theorems
Objectives:
1. To find the geometric mean of two
   numbers.
2. To find missing lengths of similar right
   triangles that result when an altitude is
   drawn to the hypotenuse
Altitudes
Recall that an altitude
 is a segment drawn
 from a vertex such
 that it is
 perpendicular to the
 opposite side of the
 triangle.
Every triangle has
 three altitudes.
Altitudes
In a right triangle, two of these altitudes are
  the two legs of the triangle. The other one
  is drawn perpendicular to the hypotenuse.
        B
                                      AB
                         Altitudes:   BC
                                      BD
 A      D                 C
Altitudes
Notice that the third altitude creates two
 smaller right triangles. Is there something
 special about the three triangles?
       B
                                     AB
                        Altitudes:   BC
                                     BD
 A      D                C
Index Card Activity
Step 1: On your index card, draw a
  diagonal. Cut along this diagonal to
  separate your right triangles.
Index Card Activity
Step 2: On one of
  the right triangles,
  fold the paper to
  create the altitude
  from the right angle
  to the hypotenuse.
  Cut along this fold
  line to separate
  your right triangles.
Index Card Activity
Step 2: On one of
  the right triangles,
  fold the paper to
  create the altitude
  from the right angle
  to the hypotenuse.
  Cut along this fold
  line to separate
  your right triangles.
Index Card Activity
Step 2: On one of
  the right triangles,
  fold the paper to
  create the altitude
  from the right angle
  to the hypotenuse.
  Cut along this fold
  line to separate
  your right triangles.
Index Card Activity
Step 3: Notice that
  the small and
  medium triangle
  can be stacked on
  the large triangle
  so that the side
  they share is the
  third altitude of the
  large triangle.
Index Card Activity
Step 3:
                                  B
Label the angles of each
   triangle to match the
   diagram.                   A         C
Be sure to label all three
   triangles.
In fact, use 4 colored            B B
   markers to color edges
   and altitude, and label
   the back of each of them
   too.                       A   D D       C
Index Card Activity
Step 4: Arrange all
  three triangles so
  that they are        A
                           B
  nested. What
  does this                    A

  demonstrate?
                       B   D   D   B   C
  Why must this be
  true?
Index Card Activity
Step 5: Write a
  similarity
  statement           A
                          B
  involving the
  large, medium,              A

  and small
                      B   D   D     B   C
  triangles.
                   ABC ~ BDC ~ ADB
Right Triangle Similarity Theorem

If an altitude is drawn
   to the hypotenuse
   of a right triangle,
   then the two
   triangles formed are
   similar to the
   original triangle and
   to each other.
Example 1
Identify the similar
  triangles in the
  diagram.
Example 2
Find the value of x.




                       Did you get x = 12   ?
                                        5
Geometric Mean
The geometric mean of two positive numbers
 a and b is the positive number x that
 satisfies        a x
                    
                  x b
              So x  ab
                  2




              And x  ab
  This is just the square root of their product!
Example 3
Find the geometric mean of 12 and 27.
Example 4
Find the value of x.



                            x



                       12       27

 Did you get x = 18?
Example 5
The altitude to the
 hypotenuse divides                     altitude
 the hypotenuse into
 two segments.
                                    x
What is the
 relationship           Segment 1
                          12
                                        Segment 2
                                           27
 between the altitude          hypotenuse
 and these two
 segments?
Geometric Mean Theorem I
Geometric Mean (Altitude)
  Theorem
In a right triangle, the altitude       x
  from the right angle to the
  hypotenuse divides the            a        b

  hypotenuse into two
  segments.                                 a x
The length of the altitude is the            
                                            x b
  geometric mean of the
  lengths of the two segments.
Geometric Mean Theorem I
Geometric Mean (Altitude)
   Heartbeat
  Theorem
In a right triangle, the altitude       x
                                        x
  from the right angle to the
  hypotenuse divides the            a
                                    a        b
                                             b

  hypotenuse into two
  segments.                                 a x
The length of the altitude is the            
                                            x b
  geometric mean of the
  lengths of the two segments.
Example 6
Find the value of w.            (w + 9)(w + 9) = (8)(18)

                                w2 + 18w + 81 = 144
(short leg)   8         w+9
                    =            w2 + 18w - 63 = 0
(long leg)    w+9       18
                                (w + 21)(w – 3) = 0

                                   w = {-21, 3}




                              8 : (3 + 9) = (3 + 9) : 18
                                  8 : 12 = 12 : 18
                                  2:3 = 2:3√
Example 7
Find the value of x.        Boomerang!
                             (short leg)      3        x
                                                   =
                             (hypotenuse)     x        12


                3      12
                                                  x2 = 36
            x




                                           Did you get x = 6?
Geometric Mean Theorem II
Geometric Mean (Leg)                                c a
 Theorem                    a
                                                     
                                                    a x
When an altitude is drawn
                                x
 to the hypotenuse of a
                                        c
 right triangle, each leg
 is the geometric mean
                                                    c b
 between the hypotenuse                         b
                                                     
 and the segment of the                             b y
 hypotenuse that is                         y

 adjacent to the leg.               c
Geometric Mean Theorem II
Geometric Mean (Leg)
Boomerang
 Theorem                  a
                          a
                                                c a
                                                 
                                                a x
When an altitude is drawn
 to the hypotenuse of a     x
                            x


 right triangle, each leg           c
                                    c


 is the geometric mean
                                                c b
 between the hypotenuse                     b
                                                 
 and the segment of the                         b y
 hypotenuse that is                     y

 adjacent to the leg.           c
Geometric Mean Theorem II
Geometric Mean (Leg)
Boomerang
 Theorem                    a
                                                    c a
                                                     
                                                    a x
When an altitude is drawn
                                x
 to the hypotenuse of a
                                        c
 right triangle, each leg
 is the geometric mean
                                                    c b
 between the hypotenuse                         b
                                                b
                                                     
 and the segment of the                             b y
 hypotenuse that is                         y
                                            y

 adjacent to the leg.               c
                                    c
Example 8
Find the value of b.




                       Did you get b = 25   ?
                                       6
Example 9
Find the value of the variable.
1. w = 6                2. k = 4




     HEARTBEAT               Boomerang!
9.3 Assignment

P 379:
(1 – 5; 8 – 13; 16, 17)

				
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