Question

the congruent sides of an isosceles triangle are each 1 unit longer than the length of the shortest side of the triangle. the perimeter of the triangle is the same as the perimeter of a square whose side length is 2 units shorter than the length of the shortest side of the triangle. what is the length of the shortest side of the triangle? units

Explanation:

Step1: Define variables

Let the length of the shortest side of the triangle be \(x\). Then the congruent sides of the isosceles triangle have length \(x + 1\). The side - length of the square is \(x-2\).

Step2: Calculate perimeters

The perimeter of the isosceles triangle \(P_{triangle}=x+(x + 1)+(x + 1)=3x+2\). The perimeter of the square \(P_{square}=4(x - 2)=4x-8\).

Step3: Set up the equation

Since \(P_{triangle}=P_{square}\), we have the equation \(3x+2=4x - 8\).

Step4: Solve the equation

Subtract \(3x\) from both sides: \(3x+2-3x=4x-8-3x\), which gives \(2=x - 8\). Then add 8 to both sides: \(2 + 8=x-8+8\).

Answer:

\(10\)