Acute Angled Triangle

In this article, we will study about the acute-angled triangle and its property.

Introduction: – A three-sided polygon with three edges, three vertices, and three interior angles is known as a triangle. A triangle is a two-dimensional figure that is closed with three sides and has three angles.

 Triangles are divided into various sorts based on their sides and angles. Each one of the triangles has its unique characteristics.

Triangles are further classified according to their sides as follows below:

  • Equilateral Triangle: A triangle with equal length sides on all three sides.
  • Isosceles Triangle: The triangle which has two sides and they are equal.
  • Scalene Triangle: A triangle with uneven sides on all three sides.

The following is a classification of triangles based on their angles:

  • Acute Angled Triangle (AAT): A triangle with all of its inner angles smaller than 90 degrees.
  • A right-angled triangle is one in which one of the inner angles is 90 degrees.
  • An obtuse angled triangle is one in which one of the inner angles is greater than 90 degrees.

ACUTE ANGLED TRIANGLE:-

An acute triangle is one with all three internal angles smaller than 90 degrees. The acute triangle’s three internal angles are all between 0 and 90 degrees, but their sum is always 180 degrees.

Angles and sides can be used to categorize triangles. An acute triangle is one that is characterized based on the angle measurements. When all of a triangle’s inner angles are fewer than 90 degrees, the triangle is said to be acute.

The definition of an acute triangle is a triangle with all three interior angles being acute angles or less than 90 degrees. Depending on whether the triangle is equilateral, isosceles, or scalene, the sides of an acute-angled triangle might be equal or unequal. In the next section, we’ll look at the different sorts of acute triangles.

Classification of acute angled triangle:-

Triangles can be classified based on their sides and angles, and an acute angled triangle can be classified further as:-

  • Equilateral Acute Triangle: All three angles in an equilateral acute triangle are equal to 60 degrees, and all sides are equal.
  • Isosceles acute triangle:- two sides and two angles are equal in an isosceles acute triangle, and all interior angles are less than 90 degrees.
  • Scalene Acute Triangle: In a scalene acute triangle, all three sides are distinct lengths, and the three interior angles are different measurements, but they are all less than 90 degrees.

The Acute Angled Triangle’s Identification:-

The triangle is steeply angled if the square of the longest side is less than the sum of the squares of two smaller sides, according to Pythagoras’ theorem. If the lengths of the sides of a given triangle are ‘a’, ‘b’, and ‘c,’ with side ‘a’ being the longest, the triangle is steeply angled if a^2< b^2 + c^2.

Properties of acute angled triangle:-

There are a few key characteristics that help us recognise an acute triangle. Below is a list of the attributes of an acute-angled triangle.

  • All three interior angles of an acute triangle add up to 180° according to the angle sum property.
  • A triangle can’t be a right-angled triangle and an acute-angled triangle at a same time.
  • The acute triangle and an obtuse triangle cannot exist in a same triangle.
  • The acute triangle’s internal angles are always less than 90° or lie between (0° and 90°), according to the angle property.
  • The smallest side of the triangle is the side opposite the smallest angle.

Formula related to acute angled triangle:-

There are two basic formulas related to the acute triangle are:-

  • Area of the acute triangle
  • Perimeter of acute triangle

Area of acute triangle:-

The formula for calculating the area of an acute triangle is Area of triangle = (1/2)*b* h. The base and height of an acute triangle are denoted by the letters ‘b’ and ‘h’ respectively.

If all of the acute triangle’s sides are known, the area of the acute triangle can be simply determined using Heron’s formula.

Using Heron’s formula, the area of an acute triangle is equal to √S(S-a)(S-b)(S-c). S stands for the semi perimeter, which can be determined using the formula Semi-perimeter (S) = (a + b + c)/2, where a, b, and c are the triangle’s sides.

Perimeter of acute triangle:-

Perimeter of triangle = (a + b + c) is the formula for calculating the perimeter of an acute triangle, which is defined as the sum of the three sides. The sides of the acute-angled triangle are a, b, and c.

Interesting facts about the acute triangle: –

Acute Triangles Have a Lot of interesting Facts:

  • Because of the Angle Sum Property, the angles of an acute triangle add up to 180°.
  • At the same time, the triangle cannot be right-angled and sharp.
  • A triangle cannot be both obtuse and Acute simultaneously.
  • With varied side measures, the interior angles of a triangle are always less than 90°.

CONCLUSION: –

The internal angles of an acute angle triangle (also known as an acute-angled triangle) are all acute angles. Remember that an acute angle is one that is less than 90 degrees. Angles and sides can be used to categorize triangles. An acute triangle is one that is characterized based on the angle measurements. When all of a triangle’s inner angles are fewer than 90 degrees, the triangle is said to be acute.

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Frequently asked questions

Get answers to the most common queries related to the CBSE 11th Examination Preparation.

What is the acute-angled triangle?

Ans:- An acute triangle is one with all three internal angles smaller than 90 degrees.

Is it true that all isosceles triangles are acute triangles?

Ans:- No, an isosceles triangle does not always have to be acute. It could be a right-angled triangle with 90°, 45...Read full

Can a triangle be acute and right at the same time?

Ans:- A triangle can be acute or right-angled, but not both. It’s impossible to be both at the same time. When...Read full

When do we tell if a triangle is acute or not?

Ans:- A triangle is acute if all of its internal angles are fewer than 90 degrees, which indicates that all angles s...Read full