Question

the triangle shown is isosceles. find the length of each side and the perimeter. each of the two congruent sides has length (type integers or decimals.) the third side has length (type an integer or a decimal.)

Explanation:

Step1: Set up equation for congruent sides

Since the triangle is isosceles, we set the two - possible equal - side equations. Let's assume either $5n - 17=2n + 1$ or $5n-17 = n$ or $2n + 1=n$. The third equation $2n + 1=n$ gives $2n - n=-1$ or $n=-1$, and side - lengths cannot be negative, so we ignore it. Also, if $5n-17 = n$, then $5n - n=17$, $4n = 17$, $n=\frac{17}{4}=4.25$. Now solve $5n-17 = 2n + 1$.
$5n-2n=1 + 17$
$3n=18$
$n = 6$

Step2: Find the lengths of the sides

Substitute $n = 6$ into the expressions for the side - lengths.
The congruent sides: Substitute $n = 6$ into $2n + 1$, we get $2\times6+1=12 + 1=13$ cm.
The third side: Substitute $n = 6$ into $n$, we get $n = 6$ cm.

Step3: Calculate the perimeter

The perimeter $P$ of a triangle is the sum of the lengths of its sides. So $P=13+13 + 6=32$ cm.

Answer:

Length of each congruent side: 13
Length of the third side: 6
Perimeter: 32