Geometry is one of the important branches of mathematics that deals with the properties of different shapes, figures, sizes, angles, and dimensions. But among all, one major concept in geometry is the triangles and their types. Basically, depending on the measurement of the angle and the relative length of the sides, the triangles are classified into various types such as Equilateral, Obtuse, Scalene, Acute, Right, and Isosceles. Also, there is another popular triangle sub-type called acute scalene triangle that shares the properties of both acute triangle and scalene triangle. Here, in this blog, let us learn about what is an Acute Scalene triangle along with its properties and examples.
Definition of Acute Scalene Triangle
As said earlier, the acute scalene triangle is a triangle type that contains the properties of both acute and scalene triangles. So, before moving to understand about acute scalene triangle, it is necessary to know about what is an acute and scalene triangle. In general, an acute triangle is a triangle with three angles less than 90 degrees whereas a scalene triangle is a triangle that has all three sides in different lengths and measures.
In geometry, an acute scalene triangles is a type of triangle whose three sides and angles will be different in measurements. Moreover, all three angles of an acute scalene triangles will also be less than 90 degrees. Say, for instance, a triangle is said to be an acute scalene triangles, if its angles are 65°, 35°, and 80°. Due to its varying internal angles, this triangle type is commonly used in roof trusses and several other modern constructions.
Properties of Acute Scalene Triangle
Listed below are the major properties of the acute scalene triangles.
- Contains three acute angles (an angle that measures less than 90°)
- Measurement of all three sides and angles is different.
- It has three intersecting line segments.
- The sum of all three interior angles is 180°.
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Acute Scalene Triangles vs. Acute Isosceles Triangle
Would you like to know how an acute scalene triangles is different from an acute isosceles triangle? If yes, then first learn the properties of an isosceles triangle. A triangle is said to be isosceles if the measure of at least two sides of a triangle is equal.
Similar to acute scalene triangles, the acute isosceles triangles also have three acute angles. But contrary to acute scalene triangles, the acute isosceles triangles have at least two congruent sides. Congruent sides indicate that the triangle’s sides are equal in length and angle.
Know How to Draw an Acute Scalene Triangle
If you want to draw an acute scalene triangle, then follow the below-mentioned steps.
- First, draw a line segment for the triangle’s base.
- Next, on each end of that line segment, construct two acute angles, the total of which should be greater than 90 degrees.
- Finally, draw the lines through those angles to meet at a common point called the triangle’s third vertex.
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Understand How to Identify an Acute Scalene Triangles
You can easily identify an acute scalene triangle if you are aware of its basic properties. In case, you don’t know the characteristics of an acute scalene triangle, you can categorize this triangle type by using a protractor as a measurement tool.
Learn to Measure Angles Using a Protractor
A protractor is often a tool used to measure curves and angles. In order to determine whether a triangle is an acute scalene triangle or not, you can measure the angles of a triangle with the protractor. Here, let us see, how to do it.
- Place the protractor’s midpoint (middle point) on the angle’s vertex.
- Align the other side with the protractor’s “zero” line.
- Trace the degrees in the protractor where the opposite side crosses the numbers scale.
- Add up all the angles and try to figure out if the triangle is an acute scalene.
A triangle is said to be acute scalene if
- All angles are less than 90°.
- All sides are unequal.
- The Sum of all three angles is 180°.
Acute Scalene Triangle Formulas
With the acute scalene triangles formulas mentioned below, you can calculate the area and perimeter of the triangle.
Formula to Calculate the Area of Acute Scalene Triangles
The formula to determine the area of acute scalene triangles,
if base and height measurements are provided,
Area= (1/2) × b × h square units, where b is the base and h is the height of the triangle.
If all of the scalene acute triangle’s sides are specified, the area of an acute scalene triangle can be simply determined using Heron’s formula, which is shown below.
Area= √S(S−a)(S−b)(S−c) square units, where S is the semi-perimeter
The semi-perimeter can be calculated using this formula
S= (a + b + c)/2, where a, b, and c are the sides of the given triangle.
Formula to Calculate the Perimeter of Acute Scalene Triangle
The perimeter is the sum of the three sides of an acute scalene triangles. It can be calculated using the formula
Perimeter P= (a + b + c) units, where a, b, and c are the sides of the given triangle.
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Acute Scalene Triangle Examples
For your better understanding, here we shared some problems on acute scalene triangles with solutions.
Example 1
Find the area of an acute scalene triangle whose base is 12 units and height is 14 units.
Solution
b= 12 units, h= 14 units
Formula to calculate the area of an acute scalene triangle
Area= (1/2) × b × h square units
Area= (1/2) x 12 x14
= 84 square units
Example 2
In a triangle PRQ, if ∠RPQ=56° and ∠PQR=74°, then what is ∠PRQ?
Solution
According to the angle sum property of an acute scalene triangle, the sum of all three interior angles should be 180°.
∠PQR + ∠PRQ + ∠RPQ = 180°.
⇒ 74° + ∠PRQ + 56° = 180°
⇒ ∠PRQ + 130° = 180°
⇒ ∠PRQ = 50°
Example 3
What will be the length of the third side of an acute scalene triangle if its perimeter is 78 inches and the lengths of the other two sides are 32 inches and 26 inches respectively?
Solution
The perimeter of an acute scalene triangles P= (a + b + c) units
As per the question, P= 78 inches; a= 32 inches; b=26 inches; c = ?
P= a + b + c
78= 32+ 26+ c
c + 58= 78
c= 78-58
c= 20 inches
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Wrapping Up
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