Question

1. if cd = 4 and the perimeter of △abc is 23, what is the perimeter of △abe?

Explanation:

Step1: Analyze the relationship between the sides

Since \(AC = AE\) (marked as equal in the figure) and \(AD\perp CE\), then \(CD=DE = 4\) (by the property of perpendicular bisector in an isosceles triangle).
The perimeter of \(\triangle ABC\) is \(P_{\triangle ABC}=AB + BC+AC=23\).

Step2: Calculate the perimeter of \(\triangle ABE\)

The perimeter of \(\triangle ABE\) is \(P_{\triangle ABE}=AB + BE+AE\).
We know that \(BE=BC + CD+DE\) and \(AC = AE\), \(CD = DE=4\).
Substitute \(BE=BC + 8\) and \(AE = AC\) into the perimeter formula of \(\triangle ABE\):
\(P_{\triangle ABE}=AB+(BC + 8)+AE=(AB + BC+AC)+8\).
Since \(AB + BC+AC = 23\) (perimeter of \(\triangle ABC\)).

Answer:

The perimeter of \(\triangle ABE\) is \(23 + 8=31\).