# What is a Scalene Triangle - Properties and Area Calculator

By the end of this lesson, you will be able to

- define scalene triangle.
- classify different types of scalene triangle.
- analyze why scalene triangle is a generalization of other triangles.
- interpret properties of scalene triangle.
- relate scalene triangle to all other triangle.
- calculate the perimeter and area of scalene triangle.
- demonstrate examples of scalene triangle.

## Scalene Triangle

The triangle having all its sides of different lengths is a scalene triangle. Another way..., the triangle having all its angles are different in measure is called a scalene triangle.

A scalene triangle

Hence, lenghts of the sides of a scalene triangle are unequal to each other. Again, all angles of a scalene triangle are also unequal to one another.

It is a general triangle of all triangles. That is, it represents all triangles. Scalene triangle is a generalization of all other triangles. That is, it represents all other tiangles.

There is a general formula to calculate the area of scalene triangle. One can also calculate the area of all other triangles using this formula.

## Apps for finding the perimeter and area of triangle

Side a:

Side b:

Side c:

Perimeter: 14.00

Area: 7.48

Drag the yellow point and see the change.

## Perimeter of Scalene Triangle

The sum of the length of all sides of scalene triangle is called perimeter.

Let BC = a, AC = b and AB = c in a scalene △ABC. Then

Perimeter, P = (a + b + c) unit

### Scalene Triangle Example

## Area of Scalene Triangle

Suppose, BC = a, AC = b and AB = c in a scalene △ABC.

Then perimeter, 2s = a + b + c

Draw a perpendicular AD to BC i.e. AD⊥BC. Let BD = x.

∴ CD = a-x.

In right △ABD and right △ACD, one can write by Pythagorean theorem

AD^{2} = AB^{2}-BD^{2} ...... (1)

AD^{2} = AC^{2}-CD^{2} ..... (2)

From (1) and (2), it can be written as

AB^{2} - BD^{2} = AC^{2} - CD^{2}

or, c^{2} - x^{2} = b^{2} - (a-x)^{2}

or, c^{2} - x^{2} = b^{2} - a^{2} + 2ax - x^{2}

or, 2ax = c^{2} + a^{2} - b^{2}

or, x = c^{2} + a^{2} - b^{2}2a

আবার, AD^{2} = c^{2} - x^{2}

or, AD^{2} = (c + x)(c - x)

or, AD^{2} = (c + c^{2} + a^{2} - b^{2}2a ) (c - c^{2} + a^{2} - b^{2}2a )

or, AD^{2} = (2ac + c^{2} + a^{2} - b^{2}2a ) (2ac - c^{2} - a^{2} + b^{2}2a )

or, AD^{2} = { (c + a)^{2} - b^{2} } { b^{2} - (c - a)^{2} }4a^{2}

or, AD^{2} = (c + a + b) (c + a - b) (b + c - a) (b - c + a)4a^{2}

or, AD^{2} = (a + b + c) (a + b + c - 2b) (a + b + c - 2a) (a + b + c - 2c)4a^{2}

or, AD^{2} = 2s (2s - 2b) (2s - 2a) (2s - 2c)4a^{2}

or, AD^{2} = 2 . 2 . 2 . 2s (s - b) (s - a) (s - c)4a^{2}

or, AD^{2} = 4s (s - a) (s - b) (s - c)a^{2}

or, AD = √4s (s - a) (s - b) (s - c)√a^{2}

or, AD = 2a √s (s - a) (s - b) (s - c)

∴△ABC = 12 BC.AD

or, △ABC = 12 a. 2a √s (s - a) (s - b) (s - c)

∴ △ABC = √s (s - a) (s - b) (s - c)

## Properties of Scalene Triangle

- The length of three sides of a scalene triangle are different in measure.
- If the length of three sides of a scalene triangle are a,b,c unit, the perimeter of the triangle, P = (a + b + c) unit.
- If the length of three sides of a scalene triangle are a, b, c unit and half the perimeter is s unit, the area of triangle, A = √s (s - a) (s - b) (s - c) square unit.
- Three medians of a scalene triangle are not equal in measure to each other.
- If the bisector of ∠A of △ABC intersect BC at a point D and intersect circumcircle ABC at a point E, then AD
^{2}= AB.AC - BD.DC. - The point at which three medians of a triangle intersect to each other is called centroid.
- The perpendicular distances from the centroid to the three sides of a scalene triangle are unequal to each other.
- If ∠B = 120° of a scalene △ABC, then AC
^{2}= AB^{2}+ BC^{2}+ AB.BC. - Three angles of a scalene triangle are different in measure.
- The perpendicular distances from three vertices to its respective opposite side of a scalene triangle are unequal to each other.
- If the lengths of three sides of a scalene triangle are a,b,c unit, the opposite angle of side c is 90° and lengths of three medians are d,e,f unit, then 2(d
^{2}+ e^{2}+ f^{2}) = 3c^{2}. - The point at which three perpendiculars from vertices to their respective opposite side of a triangle intersect to each other is called orthocenter.
- One can determine the lengths of the medians of a scalene triangle if the lengths of the sides are given.
- For a scalene △ABC, always AB
^{2}+ AC^{2}= AP^{2}+ AQ^{2}+ 4PQ^{2}if the side BC is divided into three equal parts at the points P and Q. - In a scalene triangle, incenter, centroid, circumcenter and orthocenter are always four different points.
- For a scalene △ABC, AB
^{2}+ BC^{2}= 2(AD^{2}+ BD^{2}) is the median of side BC. - The area of the square drawn on the opposite side of the obtuse angle of an obtuse triangle is equal to the total sum of the two squares drawn on the other two sides and product of twice the area of the rectangle included by either of the two other sides and the orthogonal projection of the other side.
- The sum of any two exterior angles of a triangle is always greater than 180°.
- If the lengths of the sides of a scalene △ABC are a,b,c and the lengths of the three medians drawn to a,b,c are d,e,f respectively, then

d^{2}= 2(b^{2}+ c^{2}) - a^{2}4

e^{2}= 2(c^{2}+ a^{2}) - b^{2}4

f^{2}= 2(a^{2}+ b^{2}) - c^{2}4 - The sum of any two sides of a triangle is always greater than the third one.
- The circumcenter, centroid and orthocenter of a scalene triangle lie on a same line.
- The single point at which the three angle bisectors of a triangle intersect to each other is called the incenter.
- If ∠ACB is an obtuse angle of △ABC, then AB
^{2}> AC^{2}+ BC^{2}. - The area of a scalene triangle can be determined if the three sides are known.
- If the angle bisectors of ∠B and ∠C of a △ABC intersect to each other at a point O, then ∠BOC = 90° + 12 ∠A.
- In a scalene △ABC, the perpendicular distance from vertex A to BC is AD and the perpendicular distance from vertex B to AC is BE, then BC.CD = AC.CE.
- No medians of a scalene triangle would be perpendicular to their respective sides.
- If ∠ACB = 90° of △ABC, then AB
^{2}= AC^{2}+ BC^{2}. - The sum of three medians of a triangle is always greater than its perimeter
- If D is the midpoint of side BC of △ABC, then AB + AC > 2AD.
- If a and b are the lengths of two adjacent sides and their included angle is θ, the area of the triangle A = 12 ab sinθ
- The three ex-circle can be drawn for a scalene triangle are unequal to each other.
- For a triangle ABC if ∠ACB is an acute angle, then AB
^{2}< AC^{2}+ BC^{2}. - If a,b,c are three sides and d,e,f are three medians of a triangle, then 3(a
^{2}+ b^{2}+ c^{2}) = 4(d^{2}+ e^{2}+ f^{2}). - Twice the sum of the area of the squares drawn on the medians of a right angled triangle is equal to thrice the area of the square drawn on the hypotenuse.
- The two medians PA and QB of a scalene △PQR intersect to each other at a point O, so PQ + PR > QO + RO.
- If a side of a triangle is extended, the exterior angle so formed is greater than each of the two interior opposite angles.
- The single point at which three perpendicular bisectors of a triangle meet to each other is called circumcenter.
- For a △ABC if BE and CF are two perpendiculars to AC and AB respectively from the vertices, then △ABC : △AEF = AB
^{2}: AE^{2}. - In a △ABC, AB > AC and AD is the bisector of ∠A intersects BC at a point D, then ∠ADB is an obtuse angle.
- The differece of the lengths of any two sides of a triangle is always smaller than the third one.
- The lengths of three perpendicular drawn from vertices to respective opposite side are unequal to each other for a scalene triangle.
- From any point P lying on the circumcircle of △ABC, perpendiculars PD and PE are drawn on BC and CA respectively. If the line segment ED intersects AB at the point O, then PO is perpendicular to AB, i.e; PO ⊥ AB.
- For any triangle, centroid cuts the median from vertex to opposite side at a ratio 2:1
- In a △ABC if ∠B = 60°, then AC
^{2}= AB^{2}+ BC^{2}- AB.BC. - Any median of a scalene triangle divides the triangle into two equal area of triangle in measure.
- In △ABC, if D and E are the midpoints of AB and AC respectively and bisector of ∠B and ∠C intersects at point O, then DE ∥ BC and DE = 12 BC.
- For a scalene triangle median, perpendicular bisector, angle bisector and altitude are four different line segments.
- For a scalene right △ABC if ∠C = 90° and D is the midpoint of BC, then AB
^{2}= AD^{2}+ 3BD^{2}. - For any triangle, the distances of vertices from circumcenter are equal to each other.
- In △ABC if the perpendiculars AD,BE and CF drawn from vertices to opposite sides meet a point O, then AO.OD = BO.OE = CO.OF.
- For a scalene △PQR, three medians PA, QB and RC intersect at a point O, then PA + QB + RC < PQ + QR + PR.
- In any triangle, the area of the square drawn on the opposite side of an acute angle is equal to the squares drawn on the other two sides diminished by twice the area of the rectangle included by any one of the other sides and the orthogonal projection of the other side on that side.
- In △ABC, ∠BCA is an obtuse angle, AB is the opposite side of the obtuse angle and the sides adjacent to obtuse angle are AC and BC. If CD is the orthogonal projection of the side AC on the extended side BC, then AB
^{2}= AC^{2}+ BC^{2}+ 2BC.CD. - If the three medians of a △ABC meet at a point G, then AB
^{2}+ BC^{2}+ AC^{2}= 3(GA^{2}+ GB^{2}+ GC^{2}). - The line segment joining the midpoints of any two sides of a triangle is parallel to the third side and in length it is half
- In △ABC, ∠ACB is an acute angle, AB is the opposite side of the acute angle and the other two sides are AC and BC. If CD is the orthogonal projection of the side AC on the side BC, then AB
^{2}= AC^{2}+ BC^{2}- 2BC.CD. - The perpendicular distances from incenter to sides of a triangle are equal.
- The sum of three angles of a triangle is equal to two right angles i.e; 180°.
- The sum of the areas of the squares drawn on any two sides of a triangle is equal to twice the sum of area of the squares drawn on the median of the third side and on either half of that side.
- If a side of a triangle is extended, the exterior angle so formed is equal to the sum of the two opposite interior angles.
- The distance between the orthocenter and a vertex of a triangle is twice the perpendicular distance from the circumcenter to the opposite side of that vertex.