Question

△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)Responsesm∠A<m∠B<m∠Cm angle upper A less than m angle upper B less than m angle upper Cm∠B<m∠A<m∠Cm angle upper B less than m angle upper A less than m angle upper Cm∠C<m∠A<m∠Bm angle upper C less than m angle upper A less than m angle upper Bm∠C<m∠B<m∠A

Answer 1

To solve this problem, we start by determining the side lengths of triangle $ABC$ given the perimeter and the expressions for the side lengths.

The perimeter of triangle $ABC$ is given as 37 units, and we have:

• $AB=10$
• $CA=x+6$
• $BC=2x$

Using the perimeter formula:

$$AB+CA+BC=37$$

Substituting the side lengths:

$$10+(x+6)+(2x)=37$$

Combining like terms:

$$10+x+6+2x=37$$

$$3x+16=37$$

Now, we solve for $x$:

$$3x=37-16$$ $$3x=21$$ $$x=7$$

Next, we find the lengths of $CA$ and $BC$:

$$CA=x+6=7+6=13$$ $$BC=2x=2(7)=14$$

Now we have the side lengths:

• $AB=10$
• $CA=13$
• $BC=14$

To determine the order of the angle measures, we can use the relationships between the sides and angles in a triangle: the larger the side, the larger the opposite angle.

Given the side lengths:

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

We can order the sides from smallest to largest: $$AB<CA<BC\text{(10 < 13 < 14)}$$

Now we can determine the order of the angles:

• Since $AB$ is the smallest side, $\mathrm{\angle }C$ is the smallest angle.
• Since $CA$ is the middle side, $\mathrm{\angle }B$ is the middle angle.
• Since $BC$ is the largest side, $\mathrm{\angle }A$ is the largest angle.

Thus, we have:

$$m\mathrm{\angle }C<m\mathrm{\angle }B<m\mathrm{\angle }A$$

The correct order of the angles from least to greatest is:

$\boxed{{\displaystyle m\mathrm{\angle }C<m\mathrm{\angle }B<m\mathrm{\angle }A}}$

This matches the response option:

m∠C < m∠B < m∠A.