Mathematics

Problem

Consider

\triangle ABC

in which D is a point on AB and E is a point on AC such that DE is parallel to BC. A student writes the following statement: “In

\triangle ADE

and

\triangle ABC,

AD/AB = AE/AC

(since DE

\parallel

BC), and

\angle A

is common, therefore the triangles are similar by SAS.” a. (i) Identify the error in the student’s reasoning. b. (ii) Write a correct proof to show that

\triangle ADE

is similar to

\triangle ABC,

clearly stating the similarity criterion used. c. (iii) If

AD = 5\,\text{cm},\; AB = 15\,\text{cm},\; AC = 18\,\text{cm}

and

BC = 24\,\text{cm},

calculate the lengths of AE and DE.

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