Pregunta
Which of the following TWO statements about triangles are true?
A. There is no triangle with side lengths of 14 inches, 17 inches, and 32 inches.
B. There is exactly 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches.
C. There is more than 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches.
D. There is no triangle with the angle measures of
E. There is exactly 1 triangle with the angle measures of
F. There is more than 1 triangle with the angle measures of
A. 5 units
B. 9.1 units
C. 15 units
D. 26.5 units
E. 35 units
F. 44.2 units
A. There is no triangle with side lengths of 14 inches, 17 inches, and 32 inches.
B. There is exactly 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches.
C. There is more than 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches.
D. There is no triangle with the angle measures of
E. There is exactly 1 triangle with the angle measures of
F. There is more than 1 triangle with the angle measures of
A. 5 units
B. 9.1 units
C. 15 units
D. 26.5 units
E. 35 units
F. 44.2 units
Ask by Campbell Horton. in the United States
Jan 25,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
True Statements: A and E.
Possible Third Side Lengths: 9.1 units, 15 units, and 26.5 units.
Solución
Let’s analyze each part of your question step by step.
Part 1: True Statements About Triangles
Statements:
- A. There is no triangle with side lengths of 14 inches, 17 inches, and 32 inches.
- B. There is exactly 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches.
- C. There is more than 1 triangle with side lengths of 14 inches, 17 inches, and 32 inches.
- D. There is no triangle with the angle measures of
and . - E. There is exactly 1 triangle with the angle measures of
and . - F. There is more than 1 triangle with the angle measures of
and .
Analysis:
-
For Statements A, B, and C (Side Lengths Analysis):
- Triangle Inequality Theorem: The sum of any two side lengths of a triangle must be greater than the third side.
- Check for 14, 17, 32 inches:
-
which is less than 32. - This violates the triangle inequality.
-
- Conclusion:
- Statement A is True.
- Statements B and C are False because no triangle can be formed with these side lengths.
-
For Statements D, E, and F (Angle Measures Analysis):
- Angle Sum Property: The sum of the angles in a triangle is
. - Given angles:
and . - Sum of given angles:
. - Third angle:
. - Uniqueness: Once two angles are specified, the third is uniquely determined.
- Conclusion:
- Statement E is True.
- Statements D and F are False because exactly one unique triangle can be formed with these angle measures.
- Angle Sum Property: The sum of the angles in a triangle is
**True Statements: A and E.
Part 2: Possible Lengths for the Third Side
Given:
- Two sides of a triangle: 13 units and 22 units.
Possible Third Side (x):
- Triangle Inequality Conditions:
-
→ -
→
-
- Thus, the third side must satisfy:
Options:
- A. 5 units → Invalid (5 < 9)
- B. 9.1 units → Valid (9.1 > 9 and < 35)
- C. 15 units → Valid (15 > 9 and < 35)
- D. 26.5 units → Valid (26.5 > 9 and < 35)
- E. 35 units → Invalid (35 is not less than 35)
- F. 44.2 units → Invalid (44.2 > 35)
**Valid Options for the Third Side: B. 9.1 units, C. 15 units, and D. 26.5 units
Summary of Correct Answers:
- First Question (True Statements): A and E
- Second Question (Possible Third Side Lengths): B, C, and D
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Simplifique esta solución Bonus Knowledge
When it comes to the side lengths of a triangle, they must always adhere to the triangle inequality theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side. In this case, 14, 17, and 32 fail this test since 14 + 17 is not greater than 32. That means, statement A is true: there is no triangle with those side lengths. Meanwhile, since the angles of a triangle must sum to
For the third side of a triangle with sides of 13 and 22 units, we can again apply the triangle inequality theorem. According to this theorem, the potential third side must be greater than the difference (22 - 13 = 9) and less than the sum (13 + 22 = 35). Therefore, options A (5 units), B (9.1 units), and C (15 units) could potentially work, as they all lie within that range. However, options D (26.5 units) and E (35 units) are ruled out, since 26.5 is valid but exceeds the greatest side sum, and 35 is not valid because it equals the maximum side sum. Option F (44.2 units) also fails nit’s way too long!
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