Acute triangle

Remember that the sum of angles in a triangle is equal to 180.

In an equilateral triangle, all sides and all angles are equal to each other.

Therefore, we will calculate as follows:

$x+x+x=180$

$3x=180$

We divide both sides by 3:

$x=60$

To solve this problem, we'll follow these steps:

• Step 1: Identify that an equilateral triangle has all sides of equal length, which implies its angles are also equal.
• Step 2: Utilize the property that the sum of angles in any triangle is $180^\circ$.
• Step 3: Since each angle is equal in an equilateral triangle, divide the total sum of $180^\circ$ by 3.

Now, let's work through each step:
Step 1: In an equilateral triangle, all angles are equal in size.
Step 2: The sum of angles in any triangle is always $180^\circ$.
Step 3: Divide $180^\circ$ by 3.

Calculating $180^\circ \div 3 = 60^\circ$.

Therefore, the size of each angle in an equilateral triangle is $60^\circ$.

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

To determine if the triangle is equilateral, we need to check if all three sides of the triangle are equal.

The given side lengths are $2X$, $12 - X$, and $12 - X$.

For the triangle to be equilateral, we must have the equality:

• $2X = 12 - X$

Let's solve this equation:

$\begin{aligned} 2X &= 12 - X \\ 2X + X &= 12 \\ 3X &= 12 \\ X &= \frac{12}{3} \\ X &= 4 \end{aligned}$

Substitute $X = 4$ back into the expressions for the sides:

• $2X = 2(4) = 8$

• $12 - X = 12 - 4 = 8$

• The third side, also $12 - X = 8$.

The three calculated side lengths are $8$, $8$, and $8$.

Since all three sides are equal, the triangle is an equilateral triangle.

Therefore, the answer is Yes, the triangle is equilateral.

To determine if the triangle is an acute-angled triangle, we need to understand the nature of its angles. In an acute-angled triangle, all three angles are less than $90^\circ$. However, we do not have explicit angle measures or side lengths shown in the drawing. Instead, we assess the probable nature of the depicted triangle.

Given that an acute-angled triangle must have its largest angle smaller than $90^\circ$, comparison property of triangle sides through Pythagorean type logic suggests that an acute triangle inequality $c^2 < a^2 + b^2$ (for sides $a$, $b$, and hypotenuse $c$) must hold.

In our problem, the depiction ultimately leads us to infer the implied relations among the triangle's angles. The given solution and analysis indicate it does not meet this criterion.

Hence, the triangle in the given drawing is not an acute-angled triangle, confirming the choice: No.