Scalene triangle is Geometric form plane that has three sides with different measurements, so its three angles also have different measurements.
Read too: What is the condition for the existence of a triangle?
Summary about scalene triangle
The scalene triangle is the type of triangle which has three sides with different measurements.
The three angles of the scalene triangle also have different measurements.
The longest side of a scalene triangle is opposite the angle with the largest measurement.
The shortest side of a scalene triangle is opposite the angle with the smallest measurement.
The distance between the base and the opposite vertex is the height of the scalene triangle.
The sum of the measurements of the sides of the scalene triangle is its perimeter.
The area of the scalene triangle is half the product of the base and height.
Isosceles triangle and equilateral triangle are other classifications of triangle in relation to the sides.
Regarding the angle, the triangle can be classified as obtuse-angled, acute-angled and right-angled.
What are the characteristics and properties of the scalene triangle?
The word scalene has Greek origins: skalenes means uneven, irregular. Thus, the main characteristic of the scalene triangle is that all your sides are different. Consequently, all the measurements of its angles are also different.
An important property of the scalene triangle is that the side with the largest measurement is always opposite the largest angle. Likewise, another important property is that the side with the smallest measurement is opposite the smallest angle.
How tall is the scalene triangle?
The height of a scalene triangle is the distance between the base and the opposite vertex. Due to the characteristics of this type of triangle, there is no single way to determine the height measurement: we must use the tool that best fits each case.
A possible strategy for determining height is to view this segment as the height of a right triangle and use the Pythagorean theorem. It seems difficult? Let's look at an example!
Example:
Determine the height h in the scalene triangle ABC below.
Resolution:
Note that segment AD divides triangle ABC into two right-angled triangles: ABD and ACD. Since BC = 2, consider that BD = x It is \(DC = 2-x\). Therefore, we can use the Pythagorean theorem in triangles ABD and ACD.
In triangle ABD:
\(h^2+x^2=1.5^2\)
\(h^2=2.25-x^2\)
In the ACD triangle:
\(h^2+(2-x)^2=1^2\)
\(h^2=-3+4x-x^2\)
Note that we obtain two expressions for \(h^2\). This means that
\(2.25-x^2=-3+4x-x^2\)
\(x = 1.3125\)
Substituting the value of x found in the expression \(h^2+(2-x)^2=1^2\):
\(h^2+(2-1.3125)^2=1^2\)
\(h^2=1 - 0.47265625\)
\(h=\sqrt{0.52734375} ≅ 0.72\)
The height h of triangle ABC is approximately 0.72 cm.
What is the perimeter of the scalene triangle?
O perimeter of the scalene triangle is the sum of the measurements of its three sides.
Example:
Triangle ABC has sides with measurements AB = 20 cm, BC = 32 cm and CA = 28 cm. What is the perimeter of ABC?
Resolution:
Note that ABC is scalene, as all sides have different measurements. The perimeter of ABC is:
20cm + 32cm + 28cm = 80cm
See too: Perimeter of an equilateral triangle
What is the area of the scalene triangle?
A area of the triangle scalene is the measurement of its surface. In any triangle, including the scalene, the area is given by \(\mathbf{\frac{b × h}2}\), on what B is the measurement of the base and H is the measurement of the height of the triangle.
Example:
What is the approximate area of the triangle below, knowing that h is approximately 1 cm?
Resolution:
Note that the triangle is scalene, as all sides have different measurements.
The segment with size h is the height of the triangle, that is, the distance from the base measuring 1.5 cm to the opposite vertex. As the information about h is approximate, the area obtained will also be approximate:
\(\frac{1.5×5}2=\frac{1.5×1}2=0.75\ cm^2\)
Triangle classifications
Triangles are classified according to sides and angles. According to their sides, triangles are classified into:
Scalene triangle: It is a triangle that has three sides with different measurements.
Equilateral triangle: It is a triangle that has three sides of equal length.
Isosceles triangle: is a triangle that has two sides with equal measurements.
According to the angles, triangles are classified into:
Obtuse triangle: is a triangle that has an obtuse angle (between 90º and 180º).
Acute triangle: is a triangle that has all acute angles (below 90º).
Right triangle: is a triangle that has a right angle (90º).
The following image summarizes this information:
Solved exercises on scalene triangle
Question 1
Judge the statements below as T (true) or F (false).
I. The scalene triangle has three sides of the same size.
II. The scalene triangle has three angles with different measurements.
Resolution:
I. F
II. V
The scalene triangle is the triangle that has three sides with different measurements.
Question 2
Sabrina's land is shaped like a scalene triangle with sides measuring 30 meters, 24 meters and 12 meters. How many meters of fence should Sabrina buy to fully protect the surrounding land?
A) 12
B) 24
C) 30
D) 54
E) 66
Resolution:
Alternative E.
Sabrina must buy at least enough to cover the perimeter of the land. So, she needs:
30 + 24 + 12 = 66 meters