Acute Triangle

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Table of contents
  1. Definition of an Acute Triangle
  2. Properties of an Acute Triangle
    1. Interior Angles
    2. Side-Length Relationships
    3. Altitudes and Orthocenter
    4. Medians and Centroid
    5. Incircle or Inscribed Circle
    6. Circumcircle
  3. Types of an Acute Triangle
    1. Equilateral Triangle
    2. Acute Isosceles Triangle
    3. Acute Scalene Triangle
  4. Acute Triangle Formulas
    1. Perimeter of an Acute Triangle
    2. Area of an Acute Triangle
  5. Solved Examples on Acute Triangle
  6. FAQs on Acute Triangle

In the world of geometry, triangles serve as fundamental building blocks, forming the basis of numerous mathematical concepts and practical applications. Among the various types of triangles, the acute triangle holds a distinct place due to its unique characteristics and its relevance in diverse fields, including mathematics, engineering, and architecture. Characterized by its angles that are all less than 90°, the acute triangle unveils a plethora of fascinating properties and applications, making it a cornerstone of geometric exploration. Let's explore the properties, types, formulas and examples of the acute triangle.

Definition of an Acute Triangle: An acute triangle is a triangle where all three interior angles are acute angles, measuring less than 90°. This fundamental property distinguishes it from other types of triangles, such as right triangles or obtuse triangles.

ABC

In the figure above, all the three angles of the △ABC measure less than 90°, hence it is an acute triangle.

Properties of an Acute Triangle

The acute triangle embodies several unique properties that contribute to its significance in mathematics and beyond. Below are some characteristics and properties of an acute triangle:

  • Interior Angles

    : All three interior angles of an acute triangle are less than 90°. The sum of the measures of these angles always equals 180°.

  • Side-Length Relationships

    : Acute triangles adhere to the triangle inequality theorem, which states that the sum of the lengths of any two sides of the triangle must be greater than the length of the third side. Mathematically, this can be represented as a + b > c, b + c > a, and c + a > b, where a, b, and c are the lengths of the sides.

  • Altitudes and Orthocenter

    : In an acute triangle, the altitudes (perpendicular lines from each vertex to the opposite side) all lie within the interior of the triangle. The point where all three altitudes intersect is known as the orthocenter.

  • Medians and Centroid

    : The medians of an acute triangle (lines from each vertex to the midpoint of the opposite side) intersect at a point called the centroid. The centroid divides each median in a 2:1 ratio, with the longer segment being closer to the vertex.

  • Incircle or Inscribed Circle

    : An acute triangle can have an inscribed circle, also known as an incircle, which is a circle inside a triangle just touching all three sides of the triangle and the centre of this circle is called the incentre of the triangle

  • Circumcircle

    : The circumcircle of an acute triangle is a circle that passes through all three vertices of the triangle. The center of the circumcircle is the intersection point of the perpendicular bisectors of the sides of the triangle.

Types of an Acute Triangle

Acute triangles can be categorized as:

Equilateral Triangle

All three sides of an equilateral triangle are of equal lengths and each interior angle of this triangle measures 60°. Therefore, an equilateral triangle is always an acute triangle.

Acute Isosceles Triangle

Two sides of an acute isosceles triangle are of equal length. All the angles of this triangle are acute angles and angles opposite to equal sides measure the same.

Acute Scalene Triangle

In an acute scalene triangle all three sides and angles are different in measurements and all three angles of an acute scalene triangle are less than 90°.

Acute Triangle Formulas

Below are some of the essential formulas associated with acute triangles:

Perimeter of an Acute Triangle

The perimeter of an acute triangle is the sum of the lengths of all three sides, which can be expressed as:
P=a+b+c
where a, b and c are the lengths of the sides of an acute triangle.

Area of an Acute Triangle

The area of an acute triangle can be calculated using the formula of the area of a triangle.
heightbase
A=12×base×height

Area of an Acute Triangle by Heron’s Formula

The area of an acute triangle can also be found by using Heron’s formula which is expressed as below.
A=(s−a)(s−b)(s−c)
where s is the semi-perimeter of the acute triangle and a, b, and c are its three sides. The value of s can be found using the formula: s=a+b+c2

Solved Examples on Acute Triangle

Example 1: Find the perimeter of an acute triangle if its side lengths are 9cm, 10cm and 7cm.
Given:a=9cm, b=10cm, c=7cm∵P=a+b+c∴P=9+10+7∴P=26cm
Therefore, the perimeter of the given acute triangle is 26cm.
Example 2: Find the area of an acute triangle if it's base is 8km and the height is 6km.
Given:base=8km, height=6km∵A=12×base×height∴A=12×8×6∴A=4×6∴A=24km2
Hence, the area of the given acute triangle is 24km2.

FAQs on Acute Triangle

  1. What is an acute triangle?

    An acute triangle is a type of triangle where all three interior angles are less than 90 degrees.

  2. How do you determine if a triangle is acute?

    You can determine if a triangle is acute by measuring its angles. If all three angles measure less than 90 degrees, the triangle is acute.

  3. Can a triangle be both acute and isosceles?

    Yes, a triangle can be both acute and isosceles. In fact, an isosceles triangle can be classified as "Isosceles Acute Triangle" if its two equal angles are both less than 90 degrees.

  4. What are the properties of an acute triangle?

    Properties of acute triangles include:

    • The sum of the interior angles equals 180 degrees.
    • The longest side is opposite the largest angle, which is still less than 90 degrees.
    • The altitude from any vertex falls inside the triangle.

  5. Is it possible for an equilateral triangle to be acute?

    Yes, an equilateral triangle is a special case of an acute triangle because all three angles are equal to 60 degrees, which is less than 90 degrees.

  6. Can an acute triangle be classified as scalene?

    Yes, an acute triangle can also be scalene if all three sides are of different lengths and all angles are less than 90 degrees.

  7. What is the difference between an acute triangle and an obtuse triangle?

    The primary difference is that in an acute triangle, all angles are less than 90 degrees, while in an obtuse triangle, one angle is greater than 90 degrees.

  8. Can the Pythagorean theorem be used with acute triangles?

    While the Pythagorean theorem specifically applies to right triangles, it can help determine relationships in acute triangles by checking the sides. In an acute triangle, the square of the length of the longest side is less than the sum of the squares of the other two sides.

  9. Are all angles in an acute triangle equal?

    Not necessarily. While an equilateral triangle is acute and has all angles equal, a general acute triangle can have different angle measures, as long as they all remain under 90 degrees.

  10. How do I calculate the area of an acute triangle?

    The area of an acute triangle can be found using the formula below if base and height of the triangle is known.

    Area=12×base×height
    Alternatively, if you know the three sides of an acute triangle you can use Heron’s formula.
    A=(s−a)(s−b)(s−c)
    where
    • s is the semi-perimeter of the triangle, (s=a+b+c2)
    • a, b, and c are the three sides of the triangle.

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