Acute Angled Triangle

Triangle is a three sided-polygon with three edges, three vertices and three interior angles. In other words, a triangle is a closed two-dimensional figure with three sides and three angles. 

Triangles are classified into different types on the basis of their sides and angles. Each of them has its own individual properties.


Classification of Triangles on the Basis of Their Sides is as Follows:

  1. Equilateral Triangle: A triangle whose all the three sides are of equal length.

  2. Isosceles Triangle: A triangle whose two sides are of equal length.

  3. Scalene Triangle:  A triangle whose all three sides are unequal.


Classification of Triangles on the Basis of Their Angles is as Follows:

  1. Acute Angled Triangle: A triangle whose all interior angles are less than 90°.

  2. Right Angled Triangle: A triangle whose one of the interior angles is 90°.

  3. Obtuse Angled Triangle: A triangle whose one of the interior angles is more than 90°.

 In this article, you will learn more about Acute angled triangles like its definition, properties, the formula of perimeter and area along with some solved examples.


Definition

An acute-angled triangle or acute triangle is a triangle whose all interior angles measure less than 90° degrees. (image will be updated soon)


In the above figure, the triangle ABC is an acute-angled triangle, as each of the three angles, ∠A, ∠B and ∠C measures 80°, 30° and 70° respectively which are less than 90°.

The three angles ∠A, ∠B and ∠C of an acute-angled triangle sums to 180°. 

In the above triangle ABC, ∠A + ∠B + ∠C = 80° + 30° + 70° = 180°.


Perimeter of an Acute Angled Triangle

The perimeter of an acute triangle is the sum of the length of its sides. So, if the length of the sides of the acute triangle are a, b and c units, then its perimeter is given by:

Perimeter of an acute triangle = (a + b + c) units.


Area of an Acute Angled Triangle

The area of an acute triangle is the amount of space that it occupies in a two-dimensional surface. So, the area of an acute triangle can be calculated if the length of its base and corresponding altitude (height) is known or the length of its three sides is known or length of two sides and angle between them is given.

Hence, the area of an acute triangle = \[\frac{1}{2}\] × (base) × (height) sq. units

                                                              = \[\frac{1}{2}\] × (b) × (h) sq. units

Or,

The area of acute triangle using Heron’s formula = \[\sqrt{s(s-a)(s-b)(s-c)}\] sq. units

Where, ‘a’, ‘b’ and ‘c’ are the length of sides of the acute triangle 

 And, s = semi-perimeter of triangle = \[\frac{a+b+c}{2}\]

Or,

The area of acute triangle if length of its two sides and angle between them is given.

Area of acute triangle = \[\frac{1}{2}\] × a × b × sinC 

where ‘a’ and ‘b’ are the length of  two sides and C is the angle between them.


How to Check if a Given Triangle is Acute Angled or Not?

Method 1: If the measure of angles of the triangle are given, then check the measure of its angles. If all  the three angles of the triangle measures less than 90° degrees, then the given triangle is an acute angled triangle.

Method 2: If the length of three sides of a triangle are given, then using Pythagora's identity we can easily determine if a given triangle is acute angled or not.

According to Pythagoras identity, the triangle is acutely angled if the square of the longest side is less than the sum of the squares of two smaller sides.

Let ‘a’, ‘b’ and ‘c’ are the length of sides of a given triangle, in which side ‘a’ is longest, then the given triangle is acutely angled if a2 < b2 + c2.


Properties of an Acute Angled Triangle

  1. An acute triangle has all interior angles measured less than 90° degrees. 

  2. The side opposite to the smallest angle is the smallest side of the triangle.

  3. The square of the longest side is less than the sum of the squares of two smaller sides.

  4. The points of concurrency, Centroid, Incenter, Circumcenter and Orthocenter lie inside the triangle.


Solved Example:

Q.1. Two angles of an acute angled triangle are of measure 750 and 350. Find the measure of the third angle.

Solution: Let the third angle be ∠A and the ∠B = 750 and ∠C = 350. Then, 

By interior angle sum property of triangles,
    ∠A + ∠B + ∠C = 1800
⇒ ∠A + 750 + 350 = 1800
⇒ ∠A + 1100 = 1800
⇒ ∠A = 180 -1100
⇒ ∠A = 700

So, the measure of the third angle of the given triangle is 700.


Q.2. Check if the given triangle is acute angled, the measure of length of its sides are 5cm, 12cm and 9cm.

Solution: Given that, the measure of longest side = 15cm and the measure of other two smaller sides are 5cm and 12cm.

Squaring and adding the length of two smaller sides: (5)2 + (12)2 = 169

And, squaring the length of the longest side: (9)2 = 81.

Since, the square of the longest side is less than the sum of the squares of two smaller sides. Therefore, the given triangle is an acute angled triangle.


Q.3. Find the area of an acute angled triangle whose base is 10cm and the corresponding altitude is 8cm.

Solution: Given, the base of acute angled triangle = 10cm, and height = 8cm.

Since, the area of an acute triangle = \[\frac{1}{2}\] × (base) × (height) sq. units

                                                         = \[\frac{1}{2}\] × 10 × 8 sq. cms

                                                         = 40 sq. cms