Acute Angle Triangle

In geometry, a triangle is a closed two-dimensional plane figure with three sides and three angles. A triangle is considered as a three-sided polygon. Based on the sides and the interior angles of a triangle, a triangle can be classified into different types. According to the sides of the triangle, the triangle can be classified into three types, namely

Scalene Triangle

Isosceles Triangle

Equilateral Triangle

A triangle with no equal sides or a triangle in which all the sides of a triangle are of different length. A triangle with two equal sides and two equal angles is called an isosceles triangle. Also, two angles of the triangle are of the same measure A triangle in which all three sides are equal and each interior angle of a triangle measure 60 degrees is called the equilateral triangle

According to the interior angles of the triangle, it can be classified as three types, namely

Acute Angle Triangle

Right Angle Triangle

Obtuse Angle Triangle

A triangle which consists of three acute angles in which all the angles are less than 90 degrees A triangle in which one angle measures 90 degrees and other two angles are less than 90 degrees(acute angles) A triangle in which one angle measures above 90 degrees and the other two angles that measure less than 90 degrees.

Acute Angle Triangle Definition

Any triangle that has an acute angle as all of its interior angles is defined as an acute angle triangle. (An acute angle is an angle less than 90°)

Example: Consider \(\Delta ABC\) in the figure below. The angles formed by the intersection of lines AB, BC and CA are \(\angle ABC\), \(\angle BCA\), and \(\angle CAB\) respectively. We can see that

∠ABC = 75°

∠BCA = 65°

∠BAC = 40°

Since all the three angles are less than 90°, we can infer that \(\Delta ABC\) is an acute angle triangle or acute angled triangle.

Acute Angle Triangle Formula

The formulas to find the area and perimeter of an acute triangle is given and explained below.

The area of acute angle triangle = (1/2) x b x h square units

Where,

“b” refers to the base of the triangle

“h” refers to the height of a triangle

If the sides of the triangle are given, then apply the Heron’s formula

The area of the acute triangle =  \(A = \sqrt{S (S-a)(S-b)(S-c)}\) square units

Where S is the semi perimeter of a triangle

It can be found using the formula

S = (a+b+c)/2

The perimeter of an acute triangle is equal to the sum of the length of the sides of a triangle and it is given as

perimeter = a + b + c units

Here,

a, b, and c denotes the sides of the triangle

Acute Angle Triangle Properties

The important properties of an acute triangle are as follows:

  • The interior angles of a triangle are always less than 90° with different side measures
  • In an acute triangle, the line drawn from the base of the triangle to the opposite vertex is always perpendicular
  • A triangle has three vertices
  • The interior angles of a triangle are formed when two edges of a triangle meet.

Important Terminologies

Circumcenter

A perpendicular bisector is a segment that divides any side of a triangle into two equal parts. The intersection of perpendicular bisectors of all the three sides of an acute angled forms the circumcenter and it always lies inside the triangle.

Incenter

An angular bisector is a segment that divides any angle of a triangle into two equal parts. The intersection of angular bisectors of all the three angles of an acute angle forms the incenter and it always lies inside the triangle.

Centroid

A median of a triangle is the line that connects an apex with the midpoint of the opposite side. In acute angle, the medians intersect at the centroid of the triangle and it always lies inside the triangle.

Orthocenter

An altitude of a triangle is a line that passes through an apex of a triangle and is perpendicular to the opposite side. The three altitudes of an acute angle intersect at the orthocenter and it always lies inside the triangle.

Distance Between Orthocenter and Circumcenter

For an acute angle triangle, the distance between orthocenter and circumcenter is always less than the circumradius.

Square of the Longest Side

For an acute angle triangle, the square of the longest side will always be less than the sum of the squares of the other two sides.

Practice Questions

  1. If two angles of an acute angled triangle are 85o and 30o, what is the angular measurement of the third angle?
  2. Find the area of the triangle if the length of one side is 8 cms and the corresponding altitude is 6 cms.
  3. Construct an acute angle triangle which has a base of 7 cms and base angles 65o and 75o. Find the circumcenter and orthocenter.

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Practise This Question

Find angle x if line l is parallel to line m and line n is a transversal.