Download 14002: Proportions in a right triangle

14002:
Proportions in a right triangle
• Geometric Mean: The number b is the geometric mean between the numbers a
and c if and only if a, b, and c are positive and
a
b
=
b
c
• altitude of a triangle: The altitude of a triangle is a perpendicular line segment
from a vertex of the triangle to the line of the opposite side.
• Theorem 1: The altitude to the hypotenuse of a right triangle forms two triangles that
are similar to each other and to the original triangle.
In the following example 4ABC is a right triangle with altitude BD to hypotenuse AC.
Then by Theorem 1 we have that
4ADB ∼ 4BDC ∼ 4ABC.
B
A
D
C
• Corollary 1: The altitude to the hypotenuse of a right triangle is the geometric mean
between the segments into which it divides the hypotenuse. In other words for the preceding figure we have
AD
BD
=
.
BD
CD
NOTE: Corollary 1 follows from Theorem 1 because we know 4ADB ∼ 4BDC. Therefore, we can write the above stated proportion.
• projections: In the above figure, we refer to AD and DC as the projections of the
sides AB and BC on the hypotenuse, respectively.
2
14002: PROPORTIONS IN A RIGHT TRIANGLE
• Corollary 2: Each leg of a right triangle is the geometric mean between the hypotenuse
and its projection on the hypotenuse. In other words, for the preceding figure we have
AB
AD
=
AB
AC
and
DC
BC
=
.
BC
AC
NOTE: Corollary 2 follows from Theorem 1 because we know that 4ADB ∼ 4ABC
and 4BDC ∼ 4ABC. Therefore, we have the above stated proportions.
Example 1: Solve for x.
x
2
8
Example 2: Solve for x.
12
x
8
14002: PROPORTIONS IN A RIGHT TRIANGLE
Exercises
3
For each of the following triangles, find the length of x. Round answers to two
decimal places, where necessary. Please note that the triangles are not drawn to
scale.
1.
5
x
2
2.
x
4
9
3.
5
x
3
4
14002: PROPORTIONS IN A RIGHT TRIANGLE
4.
x
20
9
5.
x
7
7
6.
11
x
x+3