Visualizing Intersecting Circles
Three cases of circle intersections.
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Triangles are formed only when the circles intersect internally (Case 3), meaning the sum of the smaller lengths exceeds the longest length.
Understand that the sum of any two sides of a triangle must be strictly greater than the third side.
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Hook exploring lengths 10, 15, and 30.
Can you build a triangle with side lengths of 10 cm, 15 cm, and 30 cm? Let's explore this question using a simple real-world analogy before doing any math.
Formal definition of the triangle inequality.
When each length is smaller than the sum of the other two, the lengths satisfy the triangle inequality. A triangle can only exist if this mathematical condition is met.
Where c is the longest side, and a and b are the two remaining shorter sides.
Instead of testing all three possible pair combinations, simply identify the longest length in the set. If the sum of the two smaller lengths > longest length, the triangle can be built!
Three cases of circle intersections.
Triangles are formed only when the circles intersect internally (Case 3), meaning the sum of the smaller lengths exceeds the longest length.
Worked example testing multiple sets of lengths.
Problem. Check if a triangle exists for the following sets of lengths:
MCQ based on source 'Figure it out' questions.
Which of the following sets of lengths can form the side lengths of a triangle?
Open practice to find possible values for a third side.
To find all possible values for X, we must set up inequalities checking the roundabout paths against the direct paths. Guide yourself through the math below.
List the side lengths that are already provided.
Set up the inequality showing that the sum of the two known sides must be strictly greater than X.
Set up the inequality assuming 7 is the longest side. (X plus the smaller known side must be strictly greater than 7).
Combine the results of your two inequalities to find the complete valid range for X.
Provide a single valid length (with units) that satisfies the constraints.