Triangles Unit Test

Question

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Question
Two sides of a triangle are 10 mm and 7 mm. Determine the length of the third side from the choices given.(1 point)
Responses

5 mm
5 mm

1 mm
1 mm

20 mm
20 mm

2 mm
2 mm
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Answer 1

To determine the possible length of the third side of a triangle when two sides are known, we can use the triangle inequality theorem. The triangle inequality states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let’s denote the two known sides as \(a = 10\) mm and \(b = 7\) mm. Let \(c\) be the length of the third side. According to the triangle inequality theorem, we have the following conditions:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

Substituting in the values we have:

1. \(10 + 7 > c\) ⟹ \(17 > c\) ⟹ \(c < 17\)
2. \(10 + c > 7\) ⟹ \(c > -3\) (This condition is always true since a side length must be positive)
3. \(7 + c > 10\) ⟹ \(c > 3\)

Combining these inequalities, we see that:

\[ 3 < c < 17 \]

Now we can evaluate the given options:

• 5 mm: This falls within the range (3 < 5 < 17), so it is valid.
• 1 mm: This is not valid (it does not satisfy \(c > 3\)).
• 20 mm: This is not valid (it exceeds \(c < 17\)).
• 2 mm: This is not valid (it does not satisfy \(c > 3\)).

From the choices provided, the only valid length for the third side is 5 mm.