similarities in a right triangle by: samuel m. gier
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SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER
How much do you know
DRILL
SIMPLIFY THE FOLLOWING EXPRESSION.
1. 4.
+
2.
5.
3.
42
45
72
64 4
464
DRILL
Find the geometric mean between the two given numbers.
1. 6 and 8
2. 9 and 4
DRILL
Find the geometric mean between the two given numbers.
1. 6 and 8
h=
=
=
h= 4
)8(6
48
)3(16
3
DRILL
Find the geometric mean between the two given numbers.
2. 9 and 4
h=
=
h= 6
)4(9
36
REVIEW ABOUT RIGHT TRIANGLES
ABBC
A
CB
LEGS
AC
&
HYPOTENUSE
The side opposite the right angle
The perpendicular side
SIMILARITIES IN A RIGHT TRIANGLE
By: SAMUEL M. GIER
CONSIDER THIS…
State the means and the extremes in the following statement.
3:7 = 6:14
The means are 7 and 6 and the extremes are 3 and 14.
CONSIDER THIS…
State the means and the extremes in the following statement.
5:3 = 6:10
The means are 3 and 6 and the extremes are 5 and 10.
CONSIDER THIS…
State the means and the extremes in the following statement.
a:h = h:b
The means are h and the extremes are a and b.
CONSIDER THIS…
Find h.
a:h = h:b
applying the law of proportion. h² = ab
h= ab
h is the geometric mean between a & b.
THEOREM:SIMILARITIES IN A RIGHT
TRIANGLE
States that “In a right triangle, the altitude to the hypotenuse separates the triangle into two triangles each similar to the given triangle and to each other.
ILLUSTRATION
“In a right triangle (∆MOR), the altitude to the hypotenuse(OS) separates the triangle into two triangles(∆MOS & ∆SOR )each similar to the given triangle (∆MOR) and to each other.
M
S
RO
∆MOR ~ ∆MSO, ∆MOR ~ ∆OSR by AA Similarity postulate)
∆MSO~ ∆OSR by transitivity
TRY THIS OUT!
NAME ALL SIMILAR TRIANGLES
A
B
DC
∆ACD ~ ∆ABC∆ACD ~ ∆CBD∆ABC ~ ∆CBD
COROLLARY 1.
In a right triangle, the altitude to the hypotenuse is the geometric
mean of the segments into which it divides the hypotenuse
ILLUSTRATION
CB is the geometric mean between AB & BD.
A
B
DC
In the figure,
BD
CB
CB
AB
COROLLARY 2.
In a right triangle, either leg is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to it.
ILLUSTRATION
CB is the geometric mean between AB & BD.
A
B
DC
In the figure,
AD
CD
CD
DB
AD
CA
CA
AB
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