How to find the length of a scalene triangle with perimeter 19
Answers
Answer:
Step-by-step explanation:
Scalene triangles must have unique side lengths (no two sides can be the same length), so it must have a side that is longer than the other two (the longest side). The other two sides also must not be the same, so one has to be longer than the other (we can call the longer side a and the shorter side b )
I am assuming we are excluding degenerate triangles (they’re not really triangles, more like flat lines). Triangles are degenerate if the longest side is the same length as the sum of the other two sides. This basically forms a straight line.
As such, the longest side must be less than half of the perimeter. The perimeter is 18cm , so the longest side must be less than 18cm2=9cm .
As the triangle has integral (positive integer) sides and given that the longest side <9cm , the longest side must be either 8cm , 7cm , 6cm , 5cm , and so on, all the way down to 1cm .
The longest side can’t be 1cm , as if it was, the rest of the triangle must have a perimeter of 18−1=17cm . There are no solutions to a+b=17 , where a and b are positive integers less than 1 (the sides must be shorter than 1 for the 1cm side to be the longest) and a>b ( a must be longer than b ). Thus, there are no triangles that fit the definition. The same logic can apply such that we rule out the longest side being between 2cm and 6cm .
The longest side must then be either 7cm or 8cm in length. If the longest side is 7cm , the other two sides must add to 11cm . The only unique positive integer solution to a+b=11 where a>b and both are less than 7 is a=6,b=5 . This makes one triangle with side lengths 7cm , 6cm and 5cm .
We can now let the longest side be 8cm . This means the other two sides must add to 10cm . The solutions to a+b=10 where a and b are positive integers less than 8 and a>b are a=7,b=3 and a=6,b=4 . This leaves two triangles with side lengths 8cm , 7cm , 3cm and 8cm , 6cm , 4cm respectively.
So we have 3 (unique) triangles that aren’t degenerate, (8cm,7cm,3cm) , (8cm,6cm,4cm) , (7cm,6cm,5cm) .
Of course there are permutations, rotations, reflections etc to these triangles, but really they are the same triangle.
Step-by-step explanation:
Scalene Triangle:
No sides have equal length
No angles are equal
Scalene Triangle Equations
These equations apply to any type of triangle. Reduced
equations for equilateral, right and isosceles are below.
Perimeter Perimeter
Semiperimeter Semiperimeter
Area Area
Area Area
Base Base
Height
Angle Bisector of side a Angle Bisector of side a
Angle Bisector of side b Angle Bisector of side b
Angle Bisector of side c Angle Bisector of side c
Median of side a Median of side a
Median of side b Median of side b
Median of side c Median of side c
Altitude of side a Altitude of side a
Altitude of side b Altitude of side b
Altitude of side c Altitude of side c
Circumscribed Circle Radius Circumscribed Circle Radius
Inscribed Circle Radius Inscribed Circle Radius
