Question

this is the only question in this section.
question
if ( cd = 4 ) and the perimeter of ( \triangle abc ) is 23, what is the perimeter of ( \triangle abe )?

Explanation:

Step1: Analyze triangle properties

From the diagram, \(AD\perp BE\) and \(AC = AE\) (marked with equal signs), so \(AD\) is the perpendicular bisector of \(CE\), meaning \(CD = DE = 4\), so \(CE=CD + DE = 8\). Also, \(AC = AE\).

Step2: Recall perimeter of \(\triangle ABC\)

Perimeter of \(\triangle ABC\) is \(AB + BC + AC=23\).

Step3: Calculate perimeter of \(\triangle ABE\)

Perimeter of \(\triangle ABE\) is \(AB + BE + AE\). Since \(BE=BC + CE\) and \(AE = AC\), substitute \(BE\) and \(AE\): \(AB + (BC + CE)+AC=(AB + BC + AC)+CE\). We know \(AB + BC + AC = 23\) and \(CE = 8\), so perimeter is \(23+8 = 31\).

Answer:

31