△ABC has a perimeter of 37 units with the following side lengths: AB¯¯¯¯¯¯¯¯=10, CA¯¯¯¯¯¯¯¯=x+6, and BC¯¯¯¯¯¯¯¯=2x. Use the side lengths of △ABC to order its angle measures. Which of the following options orders the angles from least to greatest?(1 point) Responses m∠C<m∠A<m∠B m angle upper C less than m angle upper A less than m angle upper B m∠A<m∠B<m∠C m angle upper A less than m angle upper B less than m angle upper C m∠B<m∠A<m∠C m angle upper B less than m angle upper A less than m angle upper C m∠C<m∠B<m∠A

1 answer

To determine the order of the angles in triangle \( \triangle ABC \), we first need to find the actual lengths of the sides based on the given relationship between the sides and the perimeter.

We know:

  • \( AB = 10 \)
  • \( CA = x + 6 \)
  • \( BC = 2x \)

The perimeter of triangle \( ABC \) is given as 37 units, so we can set up the following equation:

\[ AB + CA + BC = 37 \]

Substituting the lengths we have:

\[ 10 + (x + 6) + 2x = 37 \]

Simplifying this equation:

\[ 10 + x + 6 + 2x = 37 \] \[ 3x + 16 = 37 \] \[ 3x = 21 \] \[ x = 7 \]

Now we can compute the side lengths:

  • \( CA = x + 6 = 7 + 6 = 13 \)
  • \( BC = 2x = 2 \times 7 = 14 \)

Thus, the side lengths of triangle ABC are:

  • \( AB = 10 \)
  • \( CA = 13 \)
  • \( BC = 14 \)

Next, we can order the angles by observing that in a triangle, the side lengths are connected to their opposite angles such that the largest side corresponds to the largest angle and the smallest side to the smallest angle.

  • \( AB = 10 \) (opposite \( \angle C \))
  • \( CA = 13 \) (opposite \( \angle B \))
  • \( BC = 14 \) (opposite \( \angle A \))

Ordering the sides from shortest to longest, we get:

\[ 10 < 13 < 14 \]

Corresponding the angles, we have:

\[ m\angle C < m\angle B < m\angle A \]

Thus, the order of the angles from least to greatest is:

\[ m\angle C < m\angle B < m\angle A \]

The correct answer is:

m∠C < m∠B < m∠A.