Types of Triangles: Identifying and defining elements

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

• Step 1: Identify the properties of a right triangle.
• Step 2: Identify the properties of an equilateral triangle.
• Step 3: Compare these properties to determine if a right triangle can be equilateral.

Now, let's work through each step:

Step 1: A right triangle is defined by having one angle equal to $90^\circ$.
Step 2: An equilateral triangle is defined by having all three sides of equal length and all three angles equal to $60^\circ$.
Step 3: Compare the angle measurements: A right triangle cannot have all angles $60^\circ$ because it requires one angle to be $90^\circ$. Likewise, an equilateral triangle cannot have a $90^\circ$ angle, as all its angles must be $60^\circ$.

Therefore, it is impossible for a right triangle to be equilateral, as they fundamentally differ in angle requirements.

The answer to the problem is No.

Let's note in which of the triangles angle B forms a right angle, meaning an angle of 90 degrees.

In answers C+D, we can see that angle B is smaller than 90 degrees.

In answer A, it is equal to 90 degrees.

Let's analyze the problem to understand how the angles are defined in a right triangle.

A right triangle is defined as a triangle that has one angle equal to $90^\circ$. This is known as a right angle. Because the sum of all angles in any triangle must be $180^\circ$, the two remaining angles must add up to $90^\circ$ (i.e., $180^\circ - 90^\circ$).

In a right triangle, the right angle is always present, leaving the other two angles to be less than $90^\circ$ each. These angles are called acute angles. An acute angle is an angle that is less than $90^\circ$.

To summarize, the angle types in a right triangle are:

• One angle that is $90^\circ$ (a right angle).
• Two angles that are each less than $90^\circ$ (acute angles).

Given the choices, the description "Straight, sharp" correlates to the angle types in a right triangle, as "Straight" can be associated with the $90^\circ$ angle (though it's generally called a right angle) and "Sharp" correlates with acute angles.

Therefore, the correct aspect of the other two angles in a right triangle are straight (right) and sharp (acute), which matches the correct choice.

Therefore, the solution to the problem is Straight, sharp.

To determine if the triangle in the diagram is obtuse, we will visually assess the angles:

• Step 1: Identify the angles in the diagram. The triangle has three angles, with one angle appearing between the horizontal base and the left slanted side.
• Step 2: Evaluate the angle between the base and the left side. If it opens wider than a right angle, it's considered obtuse. This angle seems to be greater than $90^\circ$, indicating obtuseness.
• Step 3: Conclude based on visual inspection. Since this key angle is greater than $90^\circ$, the triangle must be an obtuse triangle.

Therefore, the solution to the problem is Yes; the diagram does show an obtuse triangle.

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

• Step 1: Understand the definition of an obtuse triangle.
• Step 2: Analyze the visual representation of the triangle in the diagram.
• Step 3: Conclude if the triangle has an angle greater than $90^\circ$.

Now, let's work through each step:
Step 1: An obtuse triangle has one angle measuring more than $90^\circ$.
Step 2: Upon observing the given diagram, the triangle appears symmetric and evenly proportioned. Typically, such geometries suggest all angles are less than or equal to $60^\circ$.

The triangle visually does not show characteristically obtuse features like a visibly extended angle, as labeled or perceptible in the typical triangular arrangement.
Step 3: Based on our observations and deductive examination of the portrayed triangle, it seems unlikely that any angle within it exceeds $90^\circ$.

Therefore, the solution to the problem is No, the diagram does not show an obtuse triangle .

To determine if the triangle shown in the diagram is obtuse, we proceed as follows:

• Step 1: Identify that the diagram is indeed a triangle by observing the confluence of three edges forming a closed shape.
• Step 2: Appreciate the geometric arrangement of the triangle, focusing on the sides' lengths and angles visually.
• Step 3: Noticeably, the longest side of the triangle represents a noticeable tilt indicating the presence of an obtuse angle.

Based on the observation above, notably from the triangle's longest side against the base, it's clear that one angle is larger than $90^\circ$. Hence, the triangle in the diagram is indeed an obtuse triangle.

Therefore, the correct answer is Yes.

To find out whether the depicted triangle is obtuse, let's recall the definition: an obtuse triangle has one angle that measures more than $90^\circ$.

In the diagram provided, we can see a triangle formed by lines drawn from the corners of what visually exists as a right angle, delineated by perpendicular segments. The prominent line bisecting these seemingly perpendicular segments does not suggest any expansion beyond each vertical or horizontal alignment inherent in the right angle setup.

Nevertheless, observe the vertex that connects these aligned angles: their linear combination and spatial property depiction give no notice of expansion over $90^\circ$.

Analyzing the configuration directly or using the properties of straight lines and angle calculations yields no evidence for an angle exceeding $90^\circ$. Therefore, the angles shown collectively correspond to a right triangle, indirectly confirmed via its geometric balance among straight, equal line segments.

Therefore, the diagram does not illustrate any feature of an obtuse triangle.

Consequently, the answer to the question "Does the diagram show an obtuse triangle?" is No.

In order to solve this problem, we need to understand the basic properties of an isosceles triangle.

An isosceles triangle has two sides that are equal in length, often referred to as the "legs" of the triangle. The angle formed between these two equal sides, which are sometimes referred to as the "sides", is called the "vertex angle" or sometimes more colloquially as the "main angle".

When considering the vocabulary of the given multiple-choice answers, choice 2: $sides, main$ accurately fills the blanks, as the angle formed between the two equal sides can indeed be referred to as the "main angle".

Therefore, the correct answer to the problem is: $sides, main$.

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 given triangle is an equilateral triangle, we must compare the lengths of all three sides. An equilateral triangle requires all three sides to have the same length.

Given the side lengths:

• $AB = 7$
• $BC = 10$
• $CA = 9$

Step-by-step solution:

• Step 1: Compare $AB$ with $BC$. We see that $7 \neq 10$. Therefore, $AB \neq BC$.
• Step 2: Compare $AB$ with $CA$. We see that $7 \neq 9$. Therefore, $AB \neq CA$.
• Step 3: Since $AB \neq BC$ and $AB \neq CA$, the sides are not all equal.

Since the sides $AB$, $BC$, and $CA$ are not all equal, the triangle is not an equilateral triangle.

Thus, the solution to the problem is No.

To determine if the triangle is equilateral, let's check the lengths of its sides:

• The first side is $2X$.
• The second side is $2X$.
• The third side is $4X$.

For a triangle to be equilateral, all side lengths must be the same, i.e., $2X = 2X = 4X$. Clearly, $2X \neq 4X$. Therefore, the sides are not equal.

Given that not all sides are of equal length, the triangle is not equilateral.

Therefore, the correct answer is No.

To determine if the triangle is equilateral, we need to verify if all three sides are of equal length. The given side lengths are:

• $AB = 6$
• $BC = 6$
• $CA = 6$

An equilateral triangle is one in which all sides are equal. Thus, we check the equality:

$AB = BC = CA$.

Substituting the given values, we have:

• $6 = 6 = 6$

Since all three sides of the triangle are indeed equal, we conclude that the triangle is an equilateral triangle.

Therefore, the correct answer is Yes.

To solve this problem, we'll determine whether a triangle with side lengths $a$, $a-2$, and $a+1$ is scalene:

• Step 1: Verify the triangle inequality theorem.
  - Check $a + (a-2) > (a+1)$: $2a - 2 > a + 1$ simplifies to $a > 3$. - Check $(a-2) + (a+1) > a$: $(2a - 1) > a$ simplifies to $a > 1$. - Check $a + (a+1) > (a-2)$: $2a + 1 > a - 2$ simplifies to $a > -3/2$, which is always true for $a > 2$.
• Step 2: Check if all sides are different.
  - Compare $a \neq a-2$: True, always holds as $a > 2$.
  - Compare $a \neq a+1$: True, always holds.
  - Compare $a-2 \neq a+1$: True, simplifies to $a \neq 3$, which holds since $a > 3$.

All side lengths satisfy the triangle inequality and are different. Therefore, the triangle is scalene. The solution to the problem is "Yes," this is a triangle with different sides.

To determine the type of triangle based on the given side lengths, we proceed as follows:

• Simplify the expressions for the side lengths:
• The first side is $a$.
• The second side simplifies as $2a-a = a$.
• The third side is $3a$.
• Compare the lengths to determine equality:
• The first side is $a$, and the second side is also $a$.
• The third side is $3a$, which is different from the first two sides ($a$).

Since the sides $a$, $a$, and $3a$ have two sides that are equal, the triangle is not a scalene triangle, which has all sides of different lengths.

Therefore, the triangle is not a triangle with different sides (scalene triangle). The correct answer is "No".

The triangle with sides 4, 4, and 5 is not a triangle with different sides. Therefore, the answer is No.

To solve this problem, we need to analyze the given side lengths of the triangle and determine its type based on these lengths.

The side lengths provided are 8, 8, and 8.

According to the definitions of triangle types:

• An equilateral triangle has all sides equal.
• An isosceles triangle has at least two sides equal.
• A scalene triangle has all sides different.

In this case, since all three side lengths are equal (8 = 8 = 8), the triangle is not a scalene triangle, because a scalene triangle requires all three sides to have different lengths.

Therefore, the triangle with sides 8, 8, and 8 is not a scalene triangle. The answer is No.

To determine if the triangle is scalene, we need to check if all sides are different and if they satisfy the triangle inequality theorem.

• Step 1: Verify all sides are different:
  Check $41 \neq 36$, $36 \neq 42$, and $41 \neq 42$. All statements are true, indicating all sides have different lengths.
• Step 2: Check the triangle inequality theorem:
  Evaluate:
  • $41 + 36 = 77 > 42$
  • $41 + 42 = 83 > 36$
  • $36 + 42 = 78 > 41$
  All inequalities are satisfied, confirming it forms a valid triangle.

Since the triangle has all different side lengths and satisfies the triangle inequality, it is indeed a scalene triangle.

Therefore, the solution to the problem is to conclude that the triangle is scalene.

The correct choice is $\text{Yes}$.

As is known, a scalene triangle is a triangle in which each side has a different length.

According to the given information, this is indeed a triangle where each side has a different length.

To determine if the given triangle is a scalene triangle, we examine the side lengths $10$, $10$, and $7$.

A triangle is classified as scalene if all three side lengths are different. Therefore, we need to check the equality between any pairs of the given side lengths:

• Check if $10 = 10$: Yes, they are equal.
• Check if $10 = 7$: No, they are not equal.
• Check if $7 = 10$: No, they are not equal.

Since the triangle has two sides of equal length ($10$ and $10$), it does not satisfy the condition for being a scalene triangle.

In conclusion, the triangle is not a scalene triangle because two of its sides are equal.

Therefore, the solution to the problem is No.

The problem requires us to determine if the triangle with given side lengths is a scalene triangle, which means all sides must be different.

We start by verifying if these side lengths form a triangle using the triangle inequality theorem, which states for any triangle with sides $a$, $b$, and $c$:

• $a + b > c$
• $a + c > b$
• $b + c > a$

Denote the given side lengths as follows:
$a = 28$, $b = 28$, $c = 25$.

Check the triangle inequalities:
$28 + 28 = 56$ which is indeed greater than $25$.
$28 + 25 = 53$ which is greater than $28$.
$25 + 28 = 53$ which is again greater than $28$.

Since all inequalities hold, these sides indeed form a triangle.

Next, determine if it is a scalene triangle. A scalene triangle has all sides of different lengths.

In our case, $a = 28 = b$, and $c = 25$. The sides $a$ and $b$ are not distinct, hence the triangle is not scalene but isosceles.

Therefore, the triangle does not have all different sides.

Thus, the correct answer is: No.