SME
Grade 10 Math · Problem type
The triangle inequality
Use this for “Two sides of a triangle are 7 and 11. What lengths can the third side be?”
Diagram
Three sides fix the triangle; the law of cosines gives each angle.
Law of sines: a/sin A = b/sin B = c/sin C ≈ 11.6
This is an example. Type your own values to start a new problem.
a
Side a
example
b
Side b
example
low
Third side is more than
worked out (not typed)
high
Third side is less than
worked out (not typed)
Formulas
low
=
|
a
−
b
|
4 = |7 − 11|
high
=
a
+
b
18 = 7 + 11
Assumptions
•
Any two sides of a triangle add to more than the third.
•
So the third side is more than the difference of the other two and less than their sum.
•
Lengths are in any one unit.
Step-by-step
WE KNOW
a = 7, b = 11
FIND
third side is more than (low), third side is less than (high)
Step 1 · Find third side is more than (low)
Use low = |a − b|
The third side must be longer than the difference.
low = |7 − 11|
low = |−4|
low = 4
Step 2 · Find third side is less than (high)
Use high = a + b
The third side must be shorter than the sum.
high = 7 + 11
high = 18 (so 4 < third side < 18)
Check
Put the numbers back in. Both sides should match.
4 = |7 − 11| ✓
18 = 7 + 11 ✓
Topic check
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