Types of Triangles: Identifying and defining elements

The problem requires us to identify the side of a right triangle that is opposite to its right angle.
In right triangles, one of the most crucial elements to recognize is the presence of a right angle (90 degrees).
The side that is directly across or opposite the right angle is known as the hypotenuse. It is also the longest side of a right triangle.
Therefore, when asked for the side opposite the right angle in a right triangle, the correct term is the hypotenuse.

Selection from the given choices corroborates our analysis:

• Choice 1: Leg - In the context of right triangles, the "legs" are the two sides that form the right angle, not the side opposite to it.
• Choice 2: Hypotenuse - This is the correct identification for the side opposite the right angle.

Therefore, the correct answer is $\text{Hypotenuse}$.

In an isosceles triangle, there are three sides: two sides of equal length and one distinct side. Our task is to identify what the equal sides are called.

To address this, let's review the basic properties of an isosceles triangle:

• An isosceles triangle is defined as a triangle with at least two sides of equal length.
• The side that is different in length from the other two is usually called the "base" of the triangle.
• The two equal sides of an isosceles triangle are referred to as the "legs."

Therefore, each of the two equal sides in an isosceles triangle is called a "leg."

In our problem, we confirm that the correct terminology for these two equal sides is indeed "legs," distinguishing them from the "base," which is the unequal side. This aligns with both the typical definitions and properties of an isosceles triangle.

Thus, the equal sides in an isosceles triangle are known as legs.

In a right triangle, there are specific terms for the sides. The two sides that form the right angle are referred to as the legs of the triangle. To differentiate, the side opposite the right angle is called the hypotenuse, which is distinct due to being the longest side. Hence, in response to the problem, the sides forming the right angle are correctly identified as Legs.

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 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 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 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 $\text{Yes}$.

To solve this problem, we need to determine if the triangle depicted is an acute-angled triangle.

An acute-angled triangle is defined as a triangle where all internal angles are less than $90^\circ$.

Upon observing the triangle in the drawing, it appears that each of its angles is less than $90^\circ$. The shape of the triangle does not present any right angles ($90^\circ$) or angles greater than $90^\circ$.

Thus, based on the visual inspection and understanding of triangle properties, the triangle appears to be acute-angled.

Therefore, the solution to the problem is Yes, the triangle is an acute-angled triangle.

To determine whether a triangle is acute-angled, we note that all interior angles must be less than $90^\circ$. While numerical angle measures are not given, the drawing representation can be analyzed.

Consider the triangle's shape in its entirety. An acute triangle, by definition, implies each angle of the triangle measures less than $90^\circ$. Therefore:

• No angle appears to be $90^\circ$ or greater based on the shape's symmetry and proportion as drawn.
• If a right angle existed, it would visually resemble an "L" flip or similar straight form.
• Acuteness indicates a slender or symmetric appearance without any extended right-angle resemblance.

Based on these observations, the triangular drawing presents no visual evidence of existing right or obtuse angles.

Therefore, the shape corresponds best with an acute-angled triangle's properties. Conclusively, the answer to whether the triangle is acute-angled is Yes.

An acute-angled triangle is defined as a triangle where all three interior angles are less than $90^\circ$.

In examining the visual depiction of the triangle provided in the problem, we need to see if it appears to satisfy this property. The assessment relies on observing the triangle's structure shown in the drawing and noting any geometric indications suggesting angle types.

Given the information from the drawing, if all angles seem to satisfy the condition of being less than $90^\circ$, then by definition, the triangle is an acute-angled triangle.

Conclusively, the answer to whether the triangle is acute-angled based on provided visual assessment and inherent assumptions in its illustration is: Yes.

Due to the presence of the 90 degree angle symbol we can determine that this is indeed a right-angled triangle.

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.

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 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.

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.

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.

To solve this problem, we will examine the side lengths of the triangle:

• Step 1: Identify each side: $a = 8$, $b = 5$, $c = 8$.
• Step 2: Compare the sides:

$a = c$ because both sides are $8$ units. This means side $a$ is equal to side $c$.

This information indicates that the triangle is not scalene because at least two sides are equal.

Therefore, the triangle is not a triangle with all different sides. It is an isosceles triangle because it has exactly two equal sides.

The correct response to the question "Given the values of the sides of a triangle, is it a triangle with different sides?" is No.

To determine if the triangle in the diagram is isosceles, we will follow these steps:

• Step 1: Identify key components of the triangle.
• Step 2: Calculate the lengths of the triangle’s sides.
• Step 3: Compare the side lengths to see if any two are equal.

From the diagram, notice the triangle appears to be a right triangle:

• We assume the base is along the horizontal from point $A$ (the right angle at (239.132, 166.627)) to point $B$ (another corner at (1091.256, 166.627)).
• The height runs vertically from point $A$ upwards (perpendicular to base).
• Hypotenuse is the line from $B$ to the topmost point (apex) of the triangle.

Let's calculate the distances:

The calculations above fail specific resolution. Evaluating actual differences on H-plane with conceptual shows all side lengths differ, as:

• Base $AB$ is longer than a side, potentially unmatched without midpoint coordinates or visually explained data specifically given line ratios.
• Existing $AC$ equal hypothesized renders Pythagorean unresolved exceeding functional equality proof due diagram inadequacy.

Therefore, since no direct component proves equivalence, the solution yields:

No, the triangle is not isosceles.

To solve the problem of determining whether the triangle in the diagram is isosceles, we first recall that an isosceles triangle is defined by having at least two equal sides or two equal angles.

Upon examining the diagram provided, we observe the triangle visually. The problem does not provide specific side lengths or angle measures, so we base our analysis on observation. In the case of an abstract or stylized diagram, typically isosceles properties would be noted or visually apparent (equal ticks on sides, angles marked as equal, etc.).

There are no such visible indicators of equal side lengths or equal angles in the diagram provided. Without explicit indications or data, the triangle appears to have all sides and angles different.

Therefore, the triangle in the diagram is not an isosceles triangle.