Obtuse Angled Triangle, Properties of Obtuse Angled Triangle 

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Delve into the world of obtuse-angled triangles and gain insights into their mathematical properties. From angle classifications to trigonometric relationships, our comprehensive overview is your go-to resource for understanding these unique geometric shapes.

Obtuse Angled Triangle

An obtuse-angled triangle, also known as an obtuse triangle, is a type of geometric figure that has one angle measuring more than 90 degrees. In a triangle, the sum of all three interior angles is always 180 degrees. Therefore, in an obtuse-angled triangle, one angle is greater than 90 degrees, making the other two angles acute (less than 90 degrees).

Key features of an obtuse-angled triangle:

1. Angles: It has one obtuse angle (greater than 90 degrees) and two acute angles (less than 90 degrees).

2. Side lengths: The side opposite the obtuse angle is called the “hypotenuse,” and it is the longest side in the triangle.

3. Examples: If you have a triangle with angles measuring, for example, 30 degrees, 60 degrees, and 90 degrees, it would be classified as a right-angled triangle because the right angle (90 degrees) is the largest. On the other hand, if the angles measure 60 degrees, 60 degrees, and 70 degrees, it would be an obtuse-angled triangle because one angle (70 degrees) is greater than 90 degrees.

4. Classification: Triangles are broadly classified based on their angles into three categories: acute-angled (all angles are less than 90 degrees), right-angled (one angle is exactly 90 degrees), and obtuse-angled (one angle is greater than 90 degrees).

5. Theorems: The properties and theorems related to obtuse-angled triangles are derived from basic principles of geometry, including the Pythagorean theorem, which relates the side lengths of a right-angled triangle.

Understanding the properties of different types of triangles, including obtuse-angled triangles, is fundamental in geometry and has applications in various fields, including physics, engineering, and computer science.

What is an Obtuse Angle Triangle?

An obtuse triangle is a triangle with one interior angle that measures more than 90 degrees. This angle is called the obtuse angle. The other two angles in the triangle must be acute, meaning they must each measure less than 90 degrees.

Here are some of the properties of obtuse triangles:

• The sum of the interior angles of a triangle is always 180 degrees. In an obtuse triangle, the obtuse angle must be greater than 90 degrees, so the other two angles must be less than 90 degrees.
• The longest side of an obtuse triangle is opposite the obtuse angle. This is because the longest side is opposite the largest angle, and in an obtuse triangle, the obtuse angle is the largest angle.
• The sum of the squares of the two smaller sides of an obtuse triangle is less than the square of the longest side. This is known as the obtuse triangle inequality.

Here are some examples of obtuse triangles:

• A triangle with sides of 5, 12, and 13 is obtuse. This is because the longest side (13) is opposite the largest angle (the angle opposite the side of length 5).
• A triangle with angles of 70, 50, and 60 is obtuse. This is because the largest angle (70) is greater than 90 degrees.

Obtuse triangles are common in everyday life. For example, they can be found in the roofs of houses, the sails of boats, and the wings of airplanes.

Types of an Obtuse Triangles

An obtuse triangle is a triangle with one obtuse angle (greater than 90°) and two acute angles. There are two main types of obtuse triangles:

1. Scalene obtuse triangle: A scalene triangle is a triangle in which no two sides or angles are equal. This means that all three sides and angles of a scalene obtuse triangle will be different.

2. Isosceles obtuse triangle: An isosceles triangle is a triangle in which two sides and the two corresponding angles are equal. This means that two of the sides of an isosceles obtuse triangle will be equal, and the two angles opposite those sides will also be equal. The third side, which is opposite the obtuse angle, will be different from the other two sides.

Here is a table summarizing the properties of obtuse triangles:

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In addition to these two main types, there are also a few special types of obtuse triangles:

• Right obtuse triangle: A right obtuse triangle is a triangle with one right angle (90°) and one obtuse angle. This type of triangle is not technically considered an obtuse triangle, but it is often included in discussions of obtuse triangles.

• Obtuse equilateral triangle: An obtuse equilateral triangle is a triangle in which all three sides are equal and all three angles are obtuse. This type of triangle does not exist, as the sum of the angles in a triangle must be 180°, and three obtuse angles would add up to more than 180°.

Properties of Obtuse Angled Triangle

An obtuse triangle is a triangle that has one angle greater than 90 degrees and less than 180 degrees. This obtuse angle is called the obtuse angle, and the other two angles are acute angles.

Here are some of the properties of obtuse-angled triangles:

1. Angle Sum Property: The sum of the interior angles of a triangle is 180 degrees. Since one angle in an obtuse triangle is obtuse, the sum of the other two acute angles must be less than 90 degrees.

2. Side-Angle-Side (SAS) Relationship: In an obtuse triangle, the side opposite the obtuse angle is the longest side of the triangle.

3. Incenter and Centroid: The incenter and centroid of an obtuse triangle lie inside the triangle.

4. Circumcenter and Orthocenter: The circumcenter and orthocenter of an obtuse triangle lie outside the triangle.

5. Classification: An obtuse triangle can be classified as a scalene triangle, isosceles triangle, or right triangle. An obtuse triangle cannot be an equilateral triangle since an equilateral triangle has all angles equal to 60 degrees.

Here is a table summarizing the properties of obtuse-angled triangles:

Obtuse-Angled Triangle Formulas

An obtuse-angled triangle is a triangle with one obtuse angle, which is an angle that measures more than 90 degrees but less than 180 degrees. The other two angles of an obtuse-angled triangle must be acute angles, which are angles that measure less than 90 degrees.

Perimeter Formula

The perimeter of a triangle is the sum of the lengths of its sides. For an obtuse-angled triangle, the perimeter formula is:

Perimeter = a + b + c

where a, b, and c are the lengths of the sides of the triangle.

Area Formula

The area of a triangle is equal to one-half the product of the base and the height. For an obtuse-angled triangle, the area formula is:

Area = (1/2) * b * h

where b is the length of the base of the triangle and h is the height of the triangle. The height is the perpendicular distance from a vertex of the triangle to its opposite side.

Law of Cosines

The Law of Cosines is a formula that relates the sides and angles of a triangle. For an obtuse-angled triangle, the Law of Cosines is:

c² = a² + b² – 2ab * cos(C)

where a, b, and c are the lengths of the sides of the triangle, and C is the measure of the obtuse angle.

Law of Sines

The Law of Sines is a formula that relates the sides and angles of a triangle. For an obtuse-angled triangle, the Law of Sines is:

a/sin(A) = b/sin(B) = c/sin(C)

where A, B, and C are the measures of the angles of the triangle, and a, b, and c are the lengths of the sides of the triangle.

These are just a few of the formulas that can be used to solve problems involving obtuse-angled triangles. The specific formula that is used will depend on the information that is given and what is being asked for.

Some Solved Examples on Obtuse-Angled Triangle

here are some solved examples on obtuse-angled triangles:

Example 1:

Given an obtuse-angled triangle with sides of length 6, 8, and 10, find the measures of the three angles.

Solution:

We can use the Law of Cosines to solve for the measures of the angles. The Law of Cosines states that:

cos(C) = (A^2 + B^2 – C^2) / (2AB)

where A, B, and C are the lengths of the sides opposite the angles of the same name.

In this case, we want to solve for angle C. We know that A = 8, B = 10, and C = 6. Plugging these values into the Law of Cosines, we get:

cos(C) = (8^2 + 10^2 – 6^2) / (2 * 8 * 10)

cos(C) = 21/40

Taking the inverse cosine of both sides, we get:

C = arccos(21/40) ≈ 78.54 degrees

We can use the same method to solve for angles A and B. We get:

A ≈ 50.76 degrees

B ≈ 50.70 degrees

Therefore, the measures of the three angles in the obtuse-angled triangle are 78.54 degrees, 50.76 degrees, and 50.70 degrees.

Example 2:

Given an obtuse-angled triangle with an angle of 100 degrees and sides of length 12 and 15, find the length of the third side.

Solution:

We can use the Law of Sines to solve for the length of the third side. The Law of Sines states that:

A/sin(A) = B/sin(B) = C/sin(C)

where A, B, and C are the lengths of the sides opposite the angles of the same name.

In this case, we want to solve for side C. We know that A = 15, B = 12, and angle C = 100 degrees. Plugging these values into the Law of Sines, we get:

15/sin(100) = 12/sin(C)

sin(C) = 12/sin(100) * sin(15)

sin(C) ≈ 0.411

Taking the inverse sine of both sides, we get:

C ≈ 23.62 degrees

Therefore, the length of the third side in the obtuse-angled triangle is approximately 23.62 units.

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