Notes on Side Lengths & Triangles - Page I
Name_________________________
MM1G3. Students will discover, prove, and apply properties of triangles,
quadrilaterals, and other polygons.
b. Understand and use the triangle inequality, the side-angle inequality, and the exterior-angle inequality.
Suppose one wants the shortest distance from point A to
point B (see right).
A
As you have possibly heard, the shortest distance between any
two points is the length of the line segment which connects
them. Let's call this Route 1. By definition, if this is the
shortest route from point A to point B, then any other route
taken would be longer.
A
B
Route 1
B
C
Therefore, consider a route (called Route 2) which starts at
point A, then goes to some other point called C, before finally
reaching point B.
Which of the two routes (1 or 2) is a longer route from
point A to point B?
Route 2
continued
Route 2
A
Route 1
B
--------------------------------------------------------------------------------------------------------------------Let's consider the ramifications of the exercise above. As you can see, ABC has been
constructed (see the picture above).
The length of Route 1 is the same as the length of side AB .
The length of Route 2 is the same as the sum of the lengths of sides AC and CB .
It has already been stated above that the length of Route 2 is longer than the length of Route 1.
Thus, the sum of the lengths of sides AC and CB is longer than the length of side AB , or
AC  CB  AB .
However, the actual lengths of all three sides of the triangle, are unimportant. They were never
given in the problem, and, hence, it is information that does not influence the statement above.
Using all of these ideas, we get the following:
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater
than the length of the third side.
--------------------------------------------------------------------------------------------------------------------Let us pause briefly to make an important statement about notation. If you are simply referring
to a side of a triangle, then you must place the segment bar above the label. For example, you
should write side AB . On the other hand, if you are referring to the length of side AB , then the
segment bar is removed like this: AB.
The Triangle Inequality Theorem allows one to state whether or not it is possible to form a
triangle with three given side lengths, because the sum of the lengths of the two shortest sides
of the triangle must be greater than the length of the longest side.
For example, can a triangle be formed with side lengths of 7 cm, 8 cm, and 12 cm?
In order to find the answer, find the sum of the two shortest sides and make sure that it is greater
than the longest side. Since 7 + 8 > 12, it is possible to form a triangle with these side lengths.
--------------------------------------------------------------------------------------------------------------------For Questions 1-3, can a triangle be formed with the given side lengths in inches?
1.
4, 5, 11
2.
12, 14, 23
3.
5, 10, 15
--------------------------------------------------------------------------------------------------------------------Suppose a new high school is being constructed such that the commons area will be triangular in
shape. If two sides of the commons will measure 25 yards and 20 yards, and the length of each
side must be a whole number of yards, what is the longest possible length for the third side?
To answer this question, remember the goal is to make the third side as long as possible. So, let's
assume it's the longest side. This would mean the other two sides (which measure 25 yards and
20 yards) would be the shortest sides.
According to the bolded and underlined statement above, the sum of the lengths of the two
shortest sides of the triangle must be greater than the length of the longest side. So, in this
problem, that means 25 + 20 (the lengths of the shortest sides) must be greater than the unknown,
longest side. Thus, 45 yards must be greater than the longest side. Since the longest side is a
whole number yards, the longest it could possibly be is 44 yards.
--------------------------------------------------------------------------------------------------------------------Three quarterbacks named Drew, Tom, and Peyton are wanting to warm up their arms with
throws of different lengths. Drew will stand 15 yards from Tom, and Tom will stand 40 yards
from Peyton. If the men want to stand in a triangle-like formation, and they wish to be a whole
number of yards from each other, what's the longest distance that Peyton can be from Drew?
--------------------------------------------------------------------------------------------------------------------Related to the above problem, what is the shortest distance that Peyton can be from Drew?
Since the unknown distance will be short, assume it is one of the shortest sides. This means its
length plus 15 (the other short side) must be greater than 40 (the longest side).
So, this question really is: what is the smallest whole number added to 15 that is greater than 40?
The answer to this question is 26 yards.
Notes on Side Lengths & Triangles - Page II
Name_________________________
Imagine for a moment that you have wrapped one large rubber band around your hands. Hold
you arms out in front of you so that they are close together. As you start widen your arms, and
increase the angle of your arms, what would be happening to the length of the rubber band?
In the most basic sense, you have formed a triangle with your two arms and the rubber band. As
was observed, as the angle of one's arms increases, the length of the rubber band (the opposite
side) also increases. This concept is communicated in the following theorems:
--------------------------------------------------------------------------------------------------------------------The longest side of a triangle is located across from the angle with the largest measure.
The shortest side of a triangle is located across from the angle with the smallest measure.
--------------------------------------------------------------------------------------------------------------------F
In the figure to the left, suppose that DF = 6 cm, EF = 5 cm, and
DE = 8 cm. Using the theorems above, which interior angle must
have the largest measure? Which must have the smallest measure?
E
D
--------------------------------------------------------------------------------------------------------------------Based on the information in the figure below, list the three sides in order, beginning with the side
with the short length and ending with the side with the longest length.
L
35
K
100
J
--------------------------------------------------------------------------------------------------------------------Homework on Side Lengths & Triangles
1.
The ancient Egyptians decide to build a pyramid for the great Raeuberes, the pharaoh of
their land. Each side other than the base of the pyramid is a triangle. Suppose the ancient
architects decide that each of these triangles will have sides measuring 1000 feet,
1000 feet, and 2000 feet. Can they construct a pyramid with these side lengths?
--------------------------------------------------------------------------------------------------------------------2.
Suppose that RST has interior angles with the following measures: mR  68 and
mS  46 . List the three sides in order, beginning with the side with the short length
and ending with the side with the longest length.
--------------------------------------------------------------------------------------------------------------------For Questions 3-5, can a triangle be formed with the given side lengths in yards?
3.
21, 27, 46
4.
0.2, 0.7, 1.0
5.
46, 55, 101
For Questions 6-8, consider the following: in WXY , WX = 23 cm, and WY = 24 cm.
6.
Which interior angle has a larger measure: X or Y ?
7.
Suppose that side XY must also have a length that is a whole number of centimeters.
What is the smallest possible length for XY ? Hint: refer to "quarterback problem".
8.
If XY must be a whole number of centimeters, what is the largest possible length for XY ?
--------------------------------------------------------------------------------------------------------------------For Questions 9-10, consider the following: in KLM , there is an exterior angle at vertex K
which measures 89 . Draw a picture of this scenario.
9.
What is mLKM ?
10.
Suppose that L measures 45 .
A)What is mM ?
B)Which side of KLM has the shortest length?
--------------------------------------------------------------------------------------------------------------------11.
Based on the theorems given in the notes today, why must the hypotenuse of a right
triangle have the longest length of all three sides? Hint - consider the measures of the
interior angles of a right triangle.
--------------------------------------------------------------------------------------------------------------------Gary is attaching tape to the bottom of the gym floor for a game. He has two pieces of tape that
are lengths of 8 feet and 10 feet. If he needs one more piece of tape to make a triangle on the
floor, and he wants the length of it to be a whole number of feet,
12
what is the shortest possible length for the third piece?
13.
what is the longest possible length for the third piece?
--------------------------------------------------------------------------------------------------------------------1. No, because 1000 + 1000 is not greater than 2000.
2. RT , RS , ST
3. Yes
4. No
5. No
6. X
7. 2 cm
8. 46 cm
9. 91
10. A) 44 B) KL
11. The hypotenuse is the side across from the 90 degree angle (the largest angle in a right
triangle), and the longest side is always across from the angle with the biggest measure.
12. 3 feet
13. 17 feet