Math Problem Statement

9 pts

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Select the two options that could be measurements of Triangle Z.

Group of answer choices

8, 11, 14

100, 175, 250

6, 9, 11

6, 12, 15

8, 14, 20

Solution

To determine which of the options could represent the side lengths of a valid triangle, we can apply the Triangle Inequality Theorem. This theorem states that, for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let's test each set:

1. 8,11,148, 11, 14

  • 8+11=198 + 11 = 19, and 19>1419 > 14
  • 8+14=228 + 14 = 22, and 22>1122 > 11
  • 11+14=2511 + 14 = 25, and 25>825 > 8

Since all conditions are satisfied, this is a valid triangle.

2. 100,175,250100, 175, 250

  • 100+175=275100 + 175 = 275, and 275>250275 > 250
  • 100+250=350100 + 250 = 350, and 350>175350 > 175
  • 175+250=425175 + 250 = 425, and 425>100425 > 100

All conditions are satisfied, so this is a valid triangle.

3. 6,9,116, 9, 11

  • 6+9=156 + 9 = 15, and 15>1115 > 11
  • 6+11=176 + 11 = 17, and 17>917 > 9
  • 9+11=209 + 11 = 20, and 20>620 > 6

This set satisfies the conditions, so it is a valid triangle.

4. 6,12,156, 12, 15

  • 6+12=186 + 12 = 18, and 18>1518 > 15
  • 6+15=216 + 15 = 21, and 21>1221 > 12
  • 12+15=2712 + 15 = 27, and 27>627 > 6

All conditions are satisfied, so this is a valid triangle.

5. 8,14,208, 14, 20

  • 8+14=228 + 14 = 22, and 22>2022 > 20
  • 8+20=288 + 20 = 28, and 28>1428 > 14
  • 14+20=3414 + 20 = 34, and 34>834 > 8

This set satisfies the conditions, so it is also a valid triangle.

Conclusion:

The two valid triangles are:

  • 8,11,148, 11, 14
  • 6,9,116, 9, 11

Would you like further details on any of these concepts?


Here are five related questions:

  1. How does the Triangle Inequality Theorem apply to different types of triangles?
  2. What are the properties of an equilateral triangle, and how do they relate to side lengths?
  3. How would the Triangle Inequality Theorem work for right triangles?
  4. Can a triangle have side lengths that are all equal? What kind of triangle is that?
  5. How does the Pythagorean theorem relate to the side lengths of a right triangle?

Tip: Always check if side lengths satisfy the Triangle Inequality Theorem to confirm if a set of numbers can form a triangle.

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Inequality Theorem

Formulas

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Theorems

Triangle Inequality Theorem

Suitable Grade Level

Grades 8-10