Complete the table with the measures of the triangle's sides. Then calculate the sum.
C
c
Add the measures of the sides to calculate their sum.
D
d
Compare the values of the sum and the third side to determine the inequality signs.
E
e
To make a conjecture, analyze the inequalities from the previous part. Is there any similarity?
A
a
B
b
Triangle
AB
BC
AB+BC
CA
Acute
5.4
5.1
10.5
3
Obtuse
4.3
6.8
11.1
4
Right
4.7
5.8
10.5
3.5
C
c
First Table:
Triangle
BC
CA
BC+CA
AB
Acute
5.1
3
8.1
5.4
Obtuse
6.8
4
10.8
4.3
Right
5.8
3.5
9.3
4.7
Second Table:
Triangle
AB
CA
AB+CA
BC
Acute
5.4
3
8.4
5.1
Obtuse
4.3
4
8.3
6.8
Right
4.7
3.5
8.2
5.8
D
d
See solution.
E
e
The measure of the sum of two triangle sides is always greater than the measure of the third side.
Practice makes perfect
a
We are asked to draw one acute, one obtuse, and one right triangle. In order to do this, let's recall their definitions.
An acute triangle is a triangle with three acute (less than 90^(∘)) angles.
An obtuse triangle is a triangle with one obtuse angle (greater than 90^(∘)) and two acute angles.
A right triangle is a triangle in which one angle is a right angle (measures 90^(∘)).
Now we can draw three triangles to match these definitions.
b
Let's measure the lengths of the triangles' sides.
Now, using the found values, we can complete the given table.
Triangle
AB
BC
AB+BC
CA
Acute
5.4
5.1
10.5
3
Obtuse
4.3
6.8
11.1
4
Right
4.7
5.8
10.5
3.5
c
We are going to make two tables similar to the one in Part B. However, this time we will find the sum of BC and CA.
Triangle
BC
CA
BC+CA
AB
Acute
5.1
3
8.1
5.4
Obtuse
6.8
4
10.8
4.3
Right
5.8
3.5
9.3
4.7
Similarly, we can calculate the sum of AB and CA.
Triangle
AB
CA
AB+CA
BC
Acute
5.4
3
8.4
5.1
Obtuse
4.3
4
8.3
6.8
Right
4.7
3.5
8.2
5.8
d
Let's compare the value of the sum in the tables from previous parts with the length of the third side placed in the most right column. We will start with the first table.
Triangle
AB+BC
CA
Inequality
Acute
10.5
3
10.5 > 3
Obtuse
11.1
4
11.1 > 4
Right
10.5
3.5
10.5 > 3.5
Now, we can do the same with the second table.
Triangle
AB+CA
BC
Inequality
Acute
8.4
5.1
8.4 > 5.1
Obtuse
8.3
6.8
8.3 > 6.8
Right
8.2
5.8
8.2 > 5.8
Finally, we can deal with the last third table.
Triangle
AB+CA
BC
Inequality
Acute
8.4
5.1
8.4 > 5.1
Obtuse
8.3
6.8
8.3 > 6.8
Right
8.2
5.8
8.2 > 5.8
e
Analyzing the tables, we can see that in each triangle the sum of the lengths of two arbitrary sides is greater than the other side length. This allows us to make a conjecture that the measure of the sum of two triangle sides is always greater than the measure of the third side.
