Download Altitude to the Hypotenuse Notetaking Worksheet

Student Name____________________________
Period______Date________________________
Altitude to the Hypotenuse Notetaking Worksheet
Discovery Problem Number 1: Use the triangle below, where ∠1 = 53°, to find the measures of all the
numbered angles, and other angles named in the chart. Record results in the chart.
T
1
Angle
Measure
Angle
Angle
Measure
Angle
Angle
Measure
∠
53°
∠1
53°
∠4
53°
∠
∠
90°
∠2
∠3
90°
∠5
∠6
90°
Angle
H
2
5
3
4
6
G
R
37°
37°
37°
What conclusion can you draw about the two smaller triangles?
•
All corresponding angles are similar. Therefore, they are similar.
What conclusion can you draw about the two smaller triangles and the big triangle?
•
All three triangles are similar.
Justify your conclusion. AA~ Postulate
•
THEOREM: The altitude to the hypotenuse divides the triangle into two triangles that are similar to
each other and to the original triangle.
EXAMPLE(S):
(Additional examples can be provided by teacher or student, as necessary).
Discovery Problem Number 2: Use the triangle below to complete the chart to identify each variable as either
a leg, altitude to the hypotenuse, or segment of the hypotenuse.
T
Variable
x
H
m
y
a
G
R
n
x
y
a
m
n
Leg, Altitude, to the Hypotenuse
or Segment of the Hypotenuse
Segment of the Hypotenuse
Segment of the Hypotenuse
m
n
Altitude to the Hypotenuse
Leg
Leg
For the last column in the chart above, if a variable is a segment
of the hypotenuse, identify to which leg it is adjacent.
•
Write a similarity statement containing three triangles.
∆~∆~∆
•
Write an extended proportion using the given variables as length for all pairs of corresponding
sides of the three similar triangles. (Hint: your extended proportion should contain 9 ratios).
+
+
= =
= = =
= = =
•
What do you notice about the means of some of the proportions?
=
=
= DEFINITION: Geometric mean of a and b, is the positive number x, such that
(Teacher may also want to include a statement that to solve for x, use:
= .
= √.
EXAMPLE(S):
(Additional examples can be provided by teacher or student, as necessary).
COROLLARY:
The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of
the segments of the hypotenuse.
EXAMPLE(S):
(From Discovery Problem Number 2,
= .
Additional examples can be provided by teacher or
student, as necessary).
COROLLARY:
The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each
leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment
of the hypotenuse adjacent to the leg.
EXAMPLE(S):
(From Discovery Problem Number 2,
=
provided by teacher or student, as necessary).
and
= .
Additional examples can be