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, there can be various types of triangles, and the acute angle triangle is one of them.
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 are of different length | A triangle with two equal sides and two equal angles is called an isosceles triangle | 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. It means that 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 measures less than 90 degrees. |
Acute Angle Triangle Definition
An acute angle triangle (or acute-angled triangle) is a triangle in which all the interior angles are acute angles. To recall, an acute angle is an angle that is less than 90°.
Example: Consider ΔABC in the figure below. The angles formed by the intersection of lines AB, BC and CA are ∠ABC, ∠BCA, and ∠CAB, respectively. We can see that,
∠ABC = ∠B = 75°
∠BCA = ∠C = 65°
∠BAC = ∠A = 40°
Since all the three angles are less than 90°, we can infer that ΔABC is an acute angle triangle or acute-angled triangle.
Acute Angle Triangle Formula
| Formulas for Acute Triangle | |
|---|---|
| Area of Acute Angle | (½) × b × h |
| The perimeter of Acute Triangle | a + b + c |
The formulas to find the area and perimeter of an acute triangle is given and explained below.
The area of acute angle triangle = (½) × b × 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.
If two sides and an interior angle is given then,
Area = (½) × ab × Sin B or,
= (½) × bc × Sin C or,
= (½) × ac × Sin A
Here, ∠A, ∠B, ∠C are the three interior angles at vertices A, B, and C, respectively. Also, a, b, and c are the lengths of sides BC, CA and AB, respectively.
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
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 triangle form 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.
Practice Questions
- If two angles of an acute-angled triangle are 85o and 30o, what is the angular measurement of the third angle?
- Find the area of the triangle if the length of one side is 8 cm and the corresponding altitude is 6 cm.
- Construct an acute angle triangle which has a base of 7 cm and base angles 65o and 75o. Find the circumcenter and orthocenter.
Frequently Asked Questions
Can an Equilateral Triangle be an Acute Angle Triangle?
Yes, all equilateral triangles are acute angle triangles. It is because an equilateral triangle has three equal angles, i.e. 60° each which are acute angles.
How to Find the Third Angle in Acute Angle Triangle?
To find the third angle of an acute triangle, add the other two sides and then subtract the sum from 180°. Thus, the formula to find the third angle is ∠A + ∠B + ∠C = 180°.
Is an Acute Scalene Triangle Possible?
Yes, an acute scalene triangle is possible if the interior angles of the scalene triangles are acute. Not only scalene, but an acute triangle can also be an isosceles triangle if it satisfies its condition.
Can a Triangle Have Only One Acute Angle?
A triangle can never have only one acute angle. If a triangle has 1 acute angle, the other angles will be either right angles or obtuse angles which is not possible as the sum of interior angles of a triangle is always 180°. So, every triangle needs to have at least 2 acute angles.
What are the Types of Triangles?
Triangles can be categorized into two main types, i.e. based on their sides or based on their interior angles. These two categories can also be further classified into various types like equilateral, scalene, acute, etc. To learn all the different types of triangles with detailed explanations, click here- https://byjus.com/maths/types-of-triangles/
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