A triangle is a three-sided polygon. There are three kinds of triangles: equilateral triangle, isosceles triangle, and scalene triangle.
A scalene triangle is a triangle that has all sides of different lengths. In other words, we can say that a scalene triangle has unequal sides or the sides of a scalene triangle are not congruent.
Following is the image showing all three kinds of triangles.
Equilateral triangle: All sides are equal as shown by the small line crossed on the sides.
Isosceles triangles: Two sides are equal as shown by two crossed lines.
Scalene triangle: All sides are unequal.
Also, all the angles in a scalene triangle are of different measures.
1. Acute scalene triangle
Acute scalene triangle means that each of the three angles has a measure of less than `90^\circ`. It is to be noted that none of the angles is a right angle (equal to `90^\circ`) or an obtuse angle (greater than `90^\circ`). The below figure shows an acute scalene triangle.
2. Right scalene triangle
A right scalene triangle is a triangle in which one angle has a measure of `90^\circ` whereas the other two acute angles have different measures. A right scalene triangle is shown below.
3. Obtuse scalene triangle
An obtuse scalene triangle is a triangle that has a measure of one of its angles greater than `90^\circ` whereas the other two acute angles have different measures.
Sum of angles in a triangle `= ∠A + ∠B + ∠C =180°`
Here, `∠A = 39°`, and `∠B = 65°`
Therefore, `∠C = 180 -(39+65)=76°`.
We know that the sides of the scalene triangle are not equal. The sum of all the sides of a scalene triangle is defined as the perimeter of the scalene triangle.
Suppose `a, b,` and `c` are the measures of the sides of a scalene triangle as shown in the figure below. Then, the perimeter is given by,
Perimeter `= a+b+c`.
The area of the scalene triangle can be calculated in two ways:
1. When the measure of base and height is given as shown in the image below.
The area of the scalene triangle is given in the same way as for an equilateral or right-angle triangle.
Area of the scalene triangle`=1/2 \times B\times H`
Where `B =` measure of the base of the scalene triangle
`H =` measure of the height of the scalene triangle.
Example: A triangle has a height of `7` cm and a base of `10` cm. Find the area of the triangle.
Solution: Here, `H=7` cm, `B=10` cm
Area of the triangle`=1/2 \times B\times H`
Area of the triangle`=1/2 \times 10\times 7=35text( cm)^2`.
2. When measure of all sides is given as shown in the image below.
To find out the area in this condition, Heron’s formula for the area of the triangle is used.
According to Heron’s formula
Area of triangle `=\sqrt(s(s-a)(s-b)(s-c))`
Where `s` is known as the semi perimeter and is given by `s=(a+b+c)/2`
Example: A triangle has its sides with a measure of `10` cm, `14` cm, and `8` cm. Find the area and the perimeter of the triangle.
Solution: Here, `a=10` cm, `b=14` cm, and `c=8` cm
`s=(10+14+8)/2=(32)/2=16\text( cm)`
Area of triangle `=\sqrt(16(16-10)(16-14)(16-8))=39.19\text( cm)^2`.
Characteristic | Equilateral Triangle | Isosceles Triangle | Scalene Triangle |
Side Lengths | All sides are equal. | At least two sides are equal. | All sides are unequal. |
Angle Measurements | All angles are `60°`. | Two angles are congruent. | All angles are incongruent. |
Number of Lines of Symmetry | Three lines of symmetry | Typically, one line of symmetry (if base angles are congruent) | No lines of symmetry. |
Perimeter Formula | `3a` | `2a+b` | `a+b+c` |
Area Formula | `\sqrt(3)/4 a^2` | `1/2 \times b\times h` | `\sqrt(s(s-a)(s-b)(s-c))` Where, `s=(a+b+c)/2` Also, `1/2 \times b\times H` |
Examples | Mercedes-Benz logo, regular stop signs | Flagpole with a triangular flag, many roof designs | Everyday triangles in various context |
1. Identify the correct statement for a scalene triangle.
2. The side lengths of the scalene triangle are `6` inches, `7` inches, and `11` inches. What is the area of the triangle?
3. A scalene triangle has a measure of `43°`, and `57°`. Find the other angle.
4. The side lengths of a scalene triangle are consecutive integers. If the perimeter of the triangle is `45` cm. Find all its sides.
Q`1`. Can a scalene triangle have one of its angles as an obtuse angle, but still be categorized as an acute scalene triangle?
No, it will not be called as an acute angle triangle because the condition of an acute angle triangle is that each angle of an acute angle triangle must have a measure of less than `90°`, otherwise it will be categorized as an obtuse angle scalene triangle.
Q`2`. Measures of how many angles are required to define a scalene triangle?
A triangle has three angles, so if you have the measure of two angles then you can use the angle sum property of the triangle to find the third angle.
Q`3`. What are the maximum lines of symmetry that can pass through the scalene triangle?
A scalene triangle like the equilateral and isosceles triangles cannot be divided equally into two parts because by nature the scalene triangle has different dimensions of its sides and that restricts it from having lines of symmetry.