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 the variable

Let the length of the shortest side of the triangle be \( x \) units. Then the length of each congruent side is \( x + 1 \) units.

Step2: Calculate the perimeter of the triangle

The perimeter of a triangle is the sum of its three sides. So the perimeter of the isosceles triangle \( P_{triangle}=x+(x + 1)+(x + 1)=3x + 2 \).

Step3: Determine the side length of the square

The side length of the square is \( x-2 \) units (since it is 2 units shorter than the shortest side of the triangle).

Step4: Calculate the perimeter of the square

The perimeter of a square is \( 4\times \) side length. So the perimeter of the square \( P_{square}=4(x - 2)=4x-8 \).

Step5: Set up the equation and solve for \( x \)

Since the perimeters of the triangle and the square are equal, we set \( 3x + 2=4x-8 \).
Subtract \( 3x \) from both sides: \( 2=x - 8 \).
Add 8 to both sides: \( x = 10 \).

Answer:

10