In ΔABC, DE ∥ BC with D on AB and E on AC. AD = 3 cm, DB = 2 cm and BC = 7.5 cm. Prove that ΔADE ~ ΔABC, find DE, and find the ratio of their perimeters.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. Given: DE ∥ BC, AD = 3 cm, DB = 2 cm, BC = 7.5 cm. To prove: ΔADE ~ ΔABC. To find: DE and the ratio of the perimeters.
2 · Draw the figure. Draw ΔABC with DE ∥ BC. Mark AD = 3 cm, DB = 2 cm, BC = 7.5 cm and put arrows on DE and BC.
3 · Write the formula or theorem. AA similarity criterion. Corresponding angles made by parallel lines are equal.
4 · Prove the similarity.
∠ADE = ∠ABC (corresponding angles, DE ∥ BC)
∠DAE = ∠BAC (common angle)
So ΔADE ~ ΔABC (AA)
5 · Find DE.
AB = AD + DB = 3 + 2 = 5 cm
DE/BC = AD/AB, so DE/7.5 = 3/5
DE = 7.5 × 3/5 = 4.5 cm
6 · Find the ratio of the perimeters. Perimeters of similar triangles are in the ratio of corresponding sides: AD/AB = 3/5, that is 3 : 5.
7 · Check and box the answer. ΔADE ~ ΔABC; DE = 4.5 cm; perimeters in the ratio 3 : 5.
Scoring tip: Use AB = AD + DB in the ratio, not DB. Using 3/2 instead of 3/5 is the most common slip.
