Use the image to answer the question.Triangle upper A upper B upper C is labeled. Upper A is at the top, upper B on the bottom right, and upper C on the bottom left. Sides are labeled: upper A upper C as x plus 10, upper A upper B as 3 x minus 2, upper B upper C as 2 x plus 4.Use the side lengths of △ABC to order its angle measures if the perimeter is 42 units. Which of the following options lists the angle measures in increasing order?Option #1: m∠C<m∠A<m∠BOption #2: m∠B<m∠C<m∠AOption #3: m∠A<m∠B<m∠C(1 point)

Answer 1

To solve the problem, we first need to express the side lengths \(AB\), \(BC\), and \(AC\) in terms of \(x\) and then set up the equation for the perimeter of triangle \(ABC\) which is given as 42 units.

The side lengths are given as:

• \(AC = x + 10\)
• \(AB = 3x - 2\)
• \(BC = 2x + 4\)

The perimeter \(P\) of triangle \(ABC\) is given by the sum of the side lengths:

\[ P = AC + AB + BC = (x + 10) + (3x - 2) + (2x + 4) \]

Combining the like terms:

\[ P = x + 10 + 3x - 2 + 2x + 4 = 6x + 12 \]

Since the perimeter is equal to 42, we set the equation:

\[ 6x + 12 = 42 \]

Now, solving for \(x\):

\[ 6x = 42 - 12 \] \[ 6x = 30 \] \[ x = 5 \]

Now we will substitute \(x = 5\) back into the expressions for the side lengths:

\[ AC = x + 10 = 5 + 10 = 15 \] \[ AB = 3x - 2 = 3(5) - 2 = 15 - 2 = 13 \] \[ BC = 2x + 4 = 2(5) + 4 = 10 + 4 = 14 \]

So the side lengths are:

• \(AC = 15\)
• \(AB = 13\)
• \(BC = 14\)

To compare the angles of the triangle, we use the fact that the angle opposite the longest side is the largest angle, and the angle opposite the shortest side is the smallest angle.

• \(AB = 13\), opposite to \(C\)
• \(BC = 14\), opposite to \(A\)
• \(AC = 15\), opposite to \(B\)

Ordering the angles based on the side lengths:

• Smallest side: \(AB = 13 \rightarrow m∠C\)
• Middle side: \(BC = 14 \rightarrow m∠A\)
• Largest side: \(AC = 15 \rightarrow m∠B\)

Thus, the order of angle measures is:

\[ m∠C < m∠A < m∠B \]

The correct answer is:

Option #1: m∠C < m∠A < m∠B.