Math Detours · Grade 7 · Sharpen · 11

Can three sticks make a triangle?

3, 4 and 8 cannot. The two short sides must be able to reach each other, and that one test decides every case.

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Tonight's idea

Take three sticks. Lay the longest one flat. Hinge a shorter stick to each end, and swing the two loose tips towards each other. If the tips can meet above the long stick, you have a triangle.

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3 and 4 cannot reach across 8. 5 and 6 can, with room to spare.

Try 3, 4 and 8. Even lying flat, the short sticks cover only \(3 + 4 = 7\) of the 8. There is a gap, and lifting the tips only widens it.

Try 5, 6 and 8. Flat, they would overlap, because \(5 + 6 = 11\). So the tips cross over, and they can be lifted until they just touch.

If the two short sticks add up to exactly the long one, the tips meet only when everything lies flat. That is a line, not a triangle.

The test. Three lengths make a triangle only when the two shorter ones add up to more than the longest.

You need to check only that one sum. Any sum that includes the longest stick is already bigger than the stick left over.

Watch one: 5, 7 and a third stick

Two sticks are 5 cm and 7 cm. Which third sticks work? 11 cm: \(5 + 7 = 12\), more than 11. Triangle.

12 cm: \(5 + 7 = 12\), exactly 12. Flat. No triangle.

2 cm: now 7 is the longest, and \(2 + 5 = 7\). Flat again. So the third stick must be longer than \(7 - 5 = 2\) and shorter than \(7 + 5 = 12\).

Slide three lengths and watch the tips try to meet: mathdetours.com/demos/sticks.html.

Your turn

  1. Which of these make a triangle? (a) 4, 5, 6. (b) 2, 3, 6. (c) 5, 5, 10. (d) 7, 10, 16.
  2. You have four sticks: 2 cm, 3 cm, 4 cm and 5 cm. List every way to pick three of them. Which picks make a triangle?
  3. Two sides of a triangle are 6 cm and 9 cm. The third side is a whole number of centimetres. (a) What is the shortest it can be? (b) The longest? (c) How many different lengths are possible?
  4. Maya's home is 5 km from school in a straight line. The school is 3 km from the pool in a straight line. (a) What is the greatest distance the pool could be from her home? (b) The least? (c) Draw a sketch for each. Could the pool be 1 km from her home? Explain.

Stretch: cut a straw

A straw 12 cm long is cut into three pieces, each a whole number of centimetres. The pieces are joined end to end to make a triangle. Find every triangle that can be made. Start by working out how long the longest piece is allowed to be.

Answers

  1. (a) Yes: \(4 + 5 = 9\), more than 6. (b) No: \(2 + 3 = 5\), less than 6. (c) No: \(5 + 5 = 10\) exactly, so it lies flat. (d) Yes: \(7 + 10 = 17\), more than 16.
  2. Four picks. 2, 3, 4: yes (\(5 \gt 4\)). 2, 3, 5: no (\(2 + 3 = 5\), flat). 2, 4, 5: yes (\(6 \gt 5\)). 3, 4, 5: yes (\(7 \gt 5\)).
  3. The third side must be longer than \(9 - 6 = 3\) and shorter than \(9 + 6 = 15\). (a) 4 cm. (b) 14 cm. (c) 4 up to 14 is 11 lengths.
  4. (a) 8 km, when home, school and pool are in a straight line with the school in the middle. (b) 2 km, when the pool lies on the straight road from school back to her home. (c) No. Home, school and pool would make a triangle with sides 1, 3 and 5, and \(1 + 3 = 4\) is less than 5. Anywhere off those two lines, the distance is between 2 km and 8 km.

Stretch. The other two pieces must add up to more than the longest, and all three add up to 12. So the longest is less than half of 12: at most 5 cm. It is also at least 4 cm, since three pieces shorter than 4 cannot reach 12. Longest 5: the others add to 7, giving 5, 5, 2 and 5, 4, 3. Longest 4: only 4, 4, 4. Three triangles.

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