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?(1 point)

Question

Responses

28 inches
28 inches
4 inches
4 inches
3 inches
3 inches
27 inches

Answer 1

To determine if Alisha can form a triangle with the two wooden pieces measuring 12 inches and 16 inches, we can use 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 third side.

Let's denote the sides of the triangle as \( a = 12 \), \( b = 16 \), and \( c \) (the length of the third piece).

The Triangle Inequality conditions can be expressed as follows:

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

Substituting the known values into these inequalities:

1. \( 12 + 16 > c \) ➔ \( 28 > c \) ➔ \( c < 28 \)
2. \( 12 + c > 16 \) ➔ \( c > 4 \)
3. \( 16 + c > 12 \) (this condition will always be true for positive \( c \))

From these inequalities, we find that \( c \) must satisfy:

\[ 4 < c < 28 \]

The possible lengths from the options given are:

• 28 inches (not valid, as it must be less than 28)
• 4 inches (not valid, as it must be greater than 4)
• 3 inches (not valid, as it must be greater than 4)
• 27 inches (valid, as it is < 28 and > 4)

Thus, the only length that could work to form a triangle is 27 inches.

Answer 2

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.(1 point)
Responses

acute
acute
scalene
scalene
isosceles
isosceles
equilateral

Answer 3

To classify the triangle with side lengths of 1 cm and 13 cm, we first need to determine the length of the third side and check if a triangle can be formed with these sides.

Using the Triangle Inequality Theorem, we know that the sum of the lengths of any two sides must be greater than the length of the third side. In this case, let's denote the third side as \( c \).

The triangle inequality conditions are:

1. \( 1 + 13 > c \) ➔ \( 14 > c \) ➔ \( c < 14 \)
2. \( 1 + c > 13 \) ➔ \( c > 12 \)
3. \( 13 + c > 1 \) (this condition will always be true for positive \( c \))

From these inequalities:

• From the first inequality, \( c < 14 \)
• From the second inequality, \( c > 12 \)

Thus, the only possible integer value for \( c \) is \( c = 13 \).

However, this means we would have side lengths of 1 cm, 13 cm, and 13 cm.

Since we have two sides of equal length (13 cm), this makes the triangle isosceles.

Now, let's confirm the classification by the angle types:

• An isosceles triangle can be acute, obtuse, or right depending on its angles. But in this case, we can also check if it fulfills the acute condition:

For a triangle to be acute, the square of the longest side must be less than the sum of the squares of the other two sides:

Let’s check:

• Longest side = 13 cm (which appears twice in this case).

So:

• \( 13^2 < 1^2 + 13^2 \)
• \( 169 < 1 + 169 \)
• \( 169 < 170 \) (True)

Thus, the triangle is isosceles and also acute.

Considering that the question specifically asks to classify by sides, the appropriate classification for this triangle is isosceles.