Transcript 5.5 - Bryan City Schools

The positions of the longest and shortest sides of
a triangle are related to the positions of the
largest and smallest angles.
Example 1A: Write the angles in order from
smallest to largest.
The shortest side is
smallest angle is F.
The longest side is
, so the
, so the largest angle is G.
The angles from smallest to largest are F, H and G.
Example 1B: Write the sides in order from
shortest to longest.
mR = 180° – (60° + 72°) = 48°
The smallest angle is R, so the
shortest side is
.
The largest angle is Q, so the longest side is
The sides from shortest to longest are
.
A triangle is formed by three segments, but not
every set of three segments can form a triangle.
A certain relationship must exist among the lengths of
three segments in order for them to form a triangle.
Example 2A: Tell whether a triangle can have sides
with the given lengths. Explain.
7, 10, 19
No—by the Triangle Inequality Theorem, a triangle
cannot have these side lengths.
Example 2B: Tell whether a triangle can have sides
with the given lengths. Explain.
2.3, 3.1, 4.6


Yes—the sum of each pair of lengths is greater
than the third length.

Example 3C: Tell whether a triangle can have
sides with the given lengths. Explain.
n + 6, n2 – 1, 3n, when n = 4.
Step 1 Evaluate each expression when n = 4.
n+6
n2 – 1
3n
4+6
(4)2 – 1
3(4)
10
15
12
Step 2 Compare the lengths.


Yes—the sum of each pair of lengths is greater
than the third length.

Example 4: Finding Side Lengths
The lengths of two sides of a triangle are 8
inches and 13 inches. Find the range of
possible lengths for the third side.
Let x represent the length of the third side. Then
apply the Triangle Inequality Theorem.
x + 8 > 13
x>5
x + 13 > 8
x > –5
8 + 13 > x
21 > x
Combine the inequalities. So 5 < x < 21. The length
of the third side is greater than 5 inches and less
than 21 inches.
Example 5: Travel Application
The figure shows the
approximate distances
between cities in California.
What is the range of distances
from San Francisco to Oakland?
Let x be the distance from San Francisco to Oakland.
x + 46 > 51 x + 51 > 46 46 + 51 > x Δ Inequal. Thm.
x>5
x > –5
97 > x
Subtr. Prop. of
Inequal.
5 < x < 97 Combine the inequalities.
The distance from San Francisco to Oakland is
greater than 5 miles and less than 97 miles.