Identification of an Isosceles Triangle

When we have a triangle, we can identify that it is an isosceles if at least one of the following conditions is met:

1) If the triangle has two equal angles - The triangle is isosceles.
2) If in the triangle the height also bisects the angle of the vertex - The triangle is isosceles.
3) If in the triangle the height is also the median - The triangle is isosceles.
4) If in the triangle the median is also the bisector - The triangle is isosceles.

Detailed explanation

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Test yourself on types of triangles!

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

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Identification of an Isosceles Triangle

Before we talk about how to identify an isosceles triangle, let's remember that it is a triangle with two sides (or edges) of the same length - This means that the base angles are also equal.
Moreover, in an isosceles triangle, the median of the base, the bisector, and the height are the same, that is, they coincide.

Let's see it illustrated

A - Identification of an isosceles triangle

These magnificent properties of the isosceles triangle cannot prove by themselves that it is an isosceles triangle.
So, how can we prove that our triangle is isosceles?

If at least one of the following conditions is met:
1) If our triangle has two equal angles - The triangle is isosceles.
This derives from the fact that the sides opposite to equal angles are also equal, therefore, if the angles are equal, the sides are too.

2) If in the triangle the height also bisects the vertex angle - The triangle is isosceles.
3) If in the triangle the height is also the median - The triangle is isosceles.
4) If in the triangle the median is also the angle bisector - The triangle is isosceles.
In fact, we can summarize guidelines $2$ and $4$ and write a single condition:
If two of these coincide - the median, the height, and the bisector - The triangle is isosceles.

Great, now you know how to identify isosceles triangles easily and quickly.

If you are interested in learning more about other angle topics, you can enter one of the following articles:

In the blog of Tutorela you will find a variety of articles about mathematics.

Examples and exercises with solutions for identifying an isosceles triangle

Exercise #1

Choose the appropriate triangle according to the following:

Angle B equals 90 degrees.

Video Solution

Step-by-Step Solution

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.

Answer

Exercise #2

Given the values of the sides of a triangle, is it a triangle with different sides?

Video Solution

Step-by-Step Solution

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.

Answer

Exercise #3

Is the triangle in the drawing an acute-angled triangle?

Video Solution

Step-by-Step Solution

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.

Answer

Yes

Exercise #4

Is the triangle in the drawing an acute-angled triangle?

Video Solution

Step-by-Step Solution

To ascertain whether the triangle in the drawing is acute, we need to examine the orientation and notation within the visual representation. The drawing vividly illustrates a triangle featuring a small square at one of the angles, a universal sign indicating a right angle. A right angle measures $90^\circ$ , rendering it impossible for the triangle to be classified as acute since an acute triangle requires all angles to be less than $90^\circ$ .

Therefore, given the right angle in the drawing, the triangle cannot be an acute-angled triangle. Consequently, the correct choice is:

No (:

)

Answer

Exercise #5

Is the triangle in the diagram isosceles?

Video Solution

Step-by-Step Solution

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:

1. **Base

AB

:** Since it's horizontal, measure the difference in x-coordinates:

AB = 1091.256 - 239.132 = 852.124

2. **Height

AC

:** This is the vertical height from point

A

to the apex which remains constant due as it stems from a vertical side.
Looks unresolved; suppose left cumulative vertical from segment width pixel movement captures well the distance that, assumably flat layout. If specifics \ say

AC = x

logically feasible, understand it scales continuous over our ground. 3. **Hypotenuse

BC

:** Since the vertex

C

sits at the vertical height same width opposite

A

against base opposite: - Using again comprehensive y-axis project addition square summed rounded hypotenuse

BC^2 = AB^2 + AC^2

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.