Alisha wants to build a miniature bridge that will include a triangle made of wooden pieces. She has two wooden pieces, one of which measures 12 inches and the other of which measures 16 inches. Which could be the length of the third piece of wood in order for her to form a triangle?
In this case, the possible lengths for the third piece of wood in order for Alisha to form a triangle are 4 and 28.
- If the third piece of wood is 4 inches, the sum of the lengths of the two shorter sides (12 inches and 4 inches) is 16 inches, which is equal to the length of the longest side (16 inches). Therefore, it is not possible to form a triangle with these lengths.
- If the third piece of wood is 28 inches, the sum of the lengths of the two shorter sides (12 inches and 28 inches) is 40 inches, which is greater than the length of the longest side (16 inches). Therefore, it is possible to form a triangle with these lengths.
Therefore, the correct option is 28 inches.
The Triangle Inequality Theorem states that in a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
Using this theorem, we can determine which types of triangles are possible with the given wooden pieces:
1) Right equilateral triangle: This type of triangle has all three sides equal in length and one right angle. With the wooden pieces measuring 12 inches and 16 inches, it is not possible to form a right equilateral triangle. In an equilateral triangle, all three sides must have the same length, but in this case, the two given sides have different lengths.
2) Obtuse equilateral triangle: This is not a valid triangle type. An equilateral triangle has all three angles equal to 60 degrees, and this cannot form an obtuse angle.
3) Equilateral scalene triangle: This type of triangle has all three sides equal in length but different angles. With the two wooden pieces measuring 12 inches and 16 inches, it is not possible to form an equilateral scalene triangle because an equilateral triangle requires all three sides to be of equal length.
4) Right isosceles triangle: This type of triangle has two sides equal in length and one right angle. With the wooden pieces measuring 12 inches and 16 inches, it is possible to form a right isosceles triangle. By using the two given sides as the legs of the triangle, the third side can be any length within the range of 4 to 28 inches.
Therefore, based on the Triangle Inequality Theorem, the only possible triangle type is a right isosceles triangle.
To classify the triangle by its sides, we need to compare the lengths of the three sides.
Given that the lengths of two sides are 1 cm and 13 cm, we can determine the possible lengths for the third side.
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
So, the possible lengths for the third side, let's call it 'x', can be described by the inequality:
1 + 13 > x
This simplifies to:
14 > x
Therefore, the length of the third side, 'x', must be less than 14 cm.
Now, let's classify the triangle by its sides:
1) If the third side 'x' is less than 12 cm, the triangle is scalene - because all three sides have different lengths.
2) If the third side 'x' is equal to 12 cm, the triangle is isosceles - because two of its sides have the same length (1 cm and 12 cm).
3) If the third side 'x' is equal to 13 cm, the triangle is degenerate - because all three of its sides have the same length (13 cm), and it essentially collapses into a single line segment.
Since the condition is that side lengths are whole numbers, we can conclude that the triangle with side lengths 1 cm, 12 cm, and 13 cm is an isosceles triangle.
To find the range of values for the third side of the triangle, we can use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let's denote the length of the third side as "x".
For the given triangle, the two given side lengths are 42.7 mm and 38.03 mm. Now, we can set up the following inequalities based on the Triangle Inequality Theorem:
x + 38.03 > 42.7 (inequality 1)
x + 42.7 > 38.03 (inequality 2)
38.03 + 42.7 > x (inequality 3)
Let's solve these inequalities:
From inequality 1:
x > 42.7 - 38.03
x > 4.67
From inequality 2:
x > 38.03 - 42.7
x > -4.67 (this inequality does not provide useful information, as length cannot be negative)
From inequality 3:
x < 38.03 + 42.7
x < 80.73
Combining the results, we find that the range of values for the third side of the triangle is:
4.67 < x < 80.73
Therefore, the third side must have a length greater than 4.67 mm and less than 80.73 mm.
To form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, for Alisha to form a triangle with the given wooden pieces, the third piece of wood must have a length that satisfies one of the following conditions:
1) The third piece of wood is shorter than 4 inches. In this case, it would not be possible to form a triangle because the sum of the lengths of the two shorter sides (12 inches and the third piece) would be less than the length of the longest side (16 inches).
2) The third piece of wood is between 4 inches and 16 inches. In this case, it would be possible to form a triangle because the sum of the lengths of the two shorter sides would be greater than the length of the longest side.
3) The third piece of wood is longer than 16 inches. In this case, it would not be possible to form a triangle because the sum of the lengths of the two shorter sides would be equal to the length of the longest side, and in a triangle, the sum of the lengths of the two shorter sides must be greater than the length of the longest side.
Therefore, the possible lengths for the third piece of wood in order for Alisha to form a triangle are between 4 inches and 16 inches.
The options are 27, 4, 3, and 28.
Based on the Triangle Inequality Theorem, which of the following types of triangles is possible?
Right equilateral
Obtuse equilateral
Equilateral scalene
Right isosceles
A triangle’s side lengths are whole numbers. If the measure of two of its sides are 1 cm and 13 cm, classify the triangle by its sides.
Find the range of values for the third side of a triangle if two of its sides measure 42.7 mm and 38.03 mm.
To determine the possible length of the third piece of wood, we can use the Triangle Inequality Theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, let's assume that the length of the third piece of wood is represented by variable 'x'. We need to check if the sum of any two sides is greater than 'x'.
For the given wooden pieces measuring 12 inches and 16 inches, we can have three possible combinations to check:
1) 12 inches + 16 inches > x
28 inches > x
2) 12 inches + x > 16 inches
x > 4 inches
3) 16 inches + x > 12 inches
x > -4 inches
Since lengths cannot be negative, the third piece of wood must have a length greater than 4 inches. Therefore, any value of 'x' greater than 4 inches would allow Alisha to form a triangle with the wooden pieces.