Chapter 3.6, Problem 16PSC

To determine

To Find: Count of isosceles triangles with sides as whole number lengths and perimeter of $18$

Answer to Problem 16PSC

The number of isosceles triangles $=\text{ 4}$

Explanation of Solution

Given information: An isosceles triangle whose sides measures in whole numbers and perimeter add up to $18$

Concept used: General rule of triangle construction possibility i.e .sum of two sides must be greater than third side is used .

Calculation: Here as per the question we have to write the different combinations for isosceles triangle where all sides are in whole numbers and perimeter is $18$

We can write all the possibilities as following:

First side	Second side	Third side	Perimeter	Verification using rule of triangles	Possibility of triangle
$1$	$1$	$16$	$18$	$1+1=2<16$	no
$2$	$2$	$14$	$18$	$2+2=4<14$	no
$3$	$3$	$12$	$18$	$3+3=6<12$	no
$4$	$4$	$10$	$18$	$4+4=8<10$	no
$5$	$5$	$8$	$18$	$5+5=10>\text{ 8}$	yes
$6$	$6$	$6$	$18$	$6+6=12>\text{ 6}$	yes
$7$	$7$	$4$	$18$	$7+7=14>\text{ 4}$	yes
$8$	$8$	$2$	$18$	$8+8=16>\text{ 2}$	yes

As shown in the above table first four cases sum of two sides is less than the third side so in those cases triangle is not possible. So, there are only four possibilities for which the required triangle can be constructed.

Conclusion:

The number of isosceles triangles $=\text{ 4}$