Question

concepts △abc is isosceles, ab = x, and bc = 2x - 4. a. find two possible values for x when the perimeter of △abc is 32. x = and b. how many possible values are there for x when the perimeter of △abc is 12? when the perimeter of △abc is 12, there is/are possible value(s) for x. the possible value(s) is/are

Explanation:

Step1: Analyze the two cases for part a

Case 1: \(AB = AC=x\).
Perimeter \(P=AB + BC+AC\), so \(32=x+(2x - 4)+x\).
Simplify the equation: \(32 = 4x-4\).
Add 4 to both sides: \(32 + 4=4x\), \(36 = 4x\).
Divide both sides by 4: \(x = 9\).
Case 2: \(BC = AC=2x - 4\).
Perimeter \(P=AB + BC+AC\), so \(32=x+(2x - 4)+(2x - 4)\).
Simplify the equation: \(32=5x-8\).
Add 8 to both sides: \(32 + 8=5x\), \(40 = 5x\).
Divide both sides by 5: \(x = 8\).

Step2: Analyze the two cases for part b

Case 1: \(AB = AC=x\).
Perimeter \(P=AB + BC+AC\), so \(12=x+(2x - 4)+x\).
Simplify the equation: \(12 = 4x-4\).
Add 4 to both sides: \(12 + 4=4x\), \(16 = 4x\).
Divide both sides by 4: \(x = 4\). Then \(BC=2x - 4=4\), and \(AB + BC>AC\) (since \(4 + 4>4\))
Case 2: \(BC = AC=2x - 4\).
Perimeter \(P=AB + BC+AC\), so \(12=x+(2x - 4)+(2x - 4)\).
Simplify the equation: \(12=5x-8\).
Add 8 to both sides: \(12 + 8=5x\), \(20 = 5x\).
Divide both sides by 5: \(x = 4\). Then \(BC=2x - 4 = 4\), \(AB=4\)

Answer:

a. \(x = 8\) and \(x = 9\)
b. There is \(1\) possible value for \(x\). The possible value is \(x = 4\)