Is Triangle ABC Obtuse-angled?.

Is Triangle ABC Obtuse-angled?

Understanding whether a triangle is obtuse-angled is a fundamental aspect of geometry that often appears in various mathematical problems and real-life applications. Triangle ABC, one of the most commonly studied triangles, can be classified based on its angles: acute, right, or obtuse. Determining if Triangle ABC is obtuse-angled involves analyzing its angles and sides using specific geometric principles and formulas. In this article, we will explore the criteria for identifying an obtuse triangle, methods to analyze Triangle ABC, and step-by-step procedures to verify if it is obtuse-angled.

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What Is an Obtuse-Angled Triangle?

Before delving into the specifics of Triangle ABC, it is essential to understand what characterizes an obtuse-angled triangle.

Definition of an Obtuse Triangle

An obtuse triangle is a triangle where one of its interior angles measures more than 90 degrees but less than 180 degrees. The key features include:

• Exactly one angle > 90°


• The other two angles are acute (< 90°)


• The side opposite the obtuse angle is the longest side in the triangle

Properties of Obtuse Triangles

• The sum of interior angles always equals 180°


• The side opposite the obtuse angle is longer than the other two sides


• The Law of Cosines is particularly useful in identifying the obtuse angle

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Methods to Determine If Triangle ABC Is Obtuse-Angled

Various methods can be employed to analyze Triangle ABC’s angles and determine whether it is obtuse. The choice of method depends on the available information about the triangle—whether you know its sides or angles.

1. Using the Lengths of Sides

The most straightforward approach involves the sides of the triangle, especially when their lengths are known.

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles:

\[ c^2 = a^2 + b^2 - 2ab \cos C \]

Similarly, for angle \( C \):

\[ \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

Procedure:

• Identify the lengths of sides \( a \), \( b \), and \( c \)


• Calculate each angle using the Law of Cosines


• Determine if any angle exceeds 90°

Criteria:

• If \(\cos C < 0\), then \( C > 90^\circ \), and the triangle is obtuse


• Repeat for all angles to confirm

2. Using the Coordinates of Vertices

When the coordinates of points A, B, and C are known, vector analysis helps determine the measure of angles.

Procedure:

• Calculate vectors \( \vec{AB} \), \( \vec{AC} \), \( \vec{BA} \), \( \vec{BC} \), etc.


• Use the dot product to find the cosine of the angles:

\[ \cos \theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} \]

• Find the angles at each vertex


• Check if any angle exceeds 90°

3. Using the Triangle’s Angles (If Known)

If the angles are given or can be measured directly:

• Verify if any angle \( > 90^\circ \)


• Confirm the triangle’s classification as obtuse

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Step-by-Step Analysis of Triangle ABC

Let’s consider a scenario where the side lengths of Triangle ABC are known:

• \( AB = 7 \) units


• \( BC = 10 \) units


• \( AC = 12 \) units

Our goal: Determine whether Triangle ABC is obtuse-angled.

Step 1: Identify the Longest Side

• The side lengths are 7, 10, and 12 units
• The longest side is \( AC = 12 \) units

Step 2: Apply the Law of Cosines to Find the Largest Angle

Since \( AC \) is the longest side, the angle opposite it is at vertex \( B \).

Using the Law of Cosines:

\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac}
\]

Where:

• \( a = AB = 7 \)


• \( b = BC = 10 \)


• \( c = AC = 12 \)

Calculate:

\[
\cos B = \frac{7^2 + 12^2 - 10^2}{2 \times 7 \times 12} = \frac{49 + 144 - 100}{168} = \frac{93}{168} \approx 0.5536
\]

Since \(\cos B \approx 0.5536\), then:

\[
B \approx \arccos(0.5536) \approx 56.6^\circ
\]

This is less than 90°, so angle \( B \) is acute.

Next, analyze the other angles:

• At vertex \( A \):

\[
\cos A = \frac{b^2 + c^2 - a^2}{2bc} = \frac{10^2 + 12^2 - 7^2}{2 \times 10 \times 12} = \frac{100 + 144 - 49}{240} = \frac{195}{240} \approx 0.8125
\]

\[
A \approx \arccos(0.8125) \approx 35.8^\circ
\]

• At vertex \( C \):

\[
\cos C = \frac{a^2 + b^2 - c^2}{2ab} = \frac{7^2 + 10^2 - 12^2}{2 \times 7 \times 10} = \frac{49 + 100 - 144}{140} = \frac{5}{140} \approx 0.0357
\]

\[
C \approx \arccos(0.0357) \approx 88^\circ
\]

Summary:

• \( A \approx 35.8^\circ \)


• \( B \approx 56.6^\circ \)


• \( C \approx 88^\circ \)

All angles are less than 90°, indicating that Triangle ABC is acute (all angles less than 90°). Therefore, Triangle ABC is not obtuse-angled.

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When Is Triangle ABC Obtuse-angled?

Based on the above methods, Triangle ABC is obtuse-angled if any one of the following conditions is met:

• Side Length Criterion: The square of the longest side is greater than the sum of the squares of the other two sides.

\[
c^2 > a^2 + b^2
\]

• Using the Law of Cosines: The cosine of any angle is negative, indicating an angle greater than 90°.

Example:

Suppose the sides of Triangle ABC are:

• \( AB = 5 \)


• \( BC = 6 \)


• \( AC = 10 \)

Check:

\[
c^2 = 10^2 = 100
\]
\[
a^2 + b^2 = 5^2 + 6^2 = 25 + 36 = 61
\]

Since \( 100 > 61 \), the triangle is obtuse-angled, with the obtuse angle opposite the side of length 10.

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Summary: How to Determine If Triangle ABC Is Obtuse-angled

Steps to analyze Triangle ABC:

1. Identify the lengths of sides \( a \), \( b \), and \( c \).


2. Determine the longest side.


3. Calculate the square of the longest side.


4. Calculate the sum of squares of the other two sides.


5. Compare:

• If \( c^2 > a^2 + b^2 \), then Triangle ABC is obtuse-angled.


• If \( c^2 = a^2 + b^2 \), then Triangle ABC is right-angled.


• If \( c^2 < a^2 + b^2 \), then Triangle ABC is acute-angled.

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Conclusion

Determining whether Triangle ABC is obtuse-angled hinges on analyzing its sides and angles using well-established geometric principles. The Law of Cosines provides a precise method to calculate angles when side lengths are known, while the side length comparison offers a quick test for obtuseness. Remember, a triangle is obtuse if and only if the square of its longest side exceeds the sum of the squares of the other two sides. By carefully applying these methods, you can confidently classify Triangle ABC and understand its geometric properties.