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Dilation

الأسبوع 9 · التمدد: مركز التمدد ومعامل التمدد والتكبير والتصغير وصورة نقطة أو مضلع في المستوى الإحداثي · 3 دروس
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The image of a triangle: finding k, proving DE ∥ BC and finding a length

Mathematics Prep 3 Term 1, Unit 3, Lesson 3
GoalUse a dilation to find lengths in a triangle, and decide whether DE is parallel to BC from the ratios.
New words
image of a triangle · the triangle formed by the images of its three verticesparallel sides · each image side is parallel to its original sideratio of lengths · image length ÷ original length = kconverse · the statement turned round: equal ratios on the sides give parallel lines
Explanation

Let D be on AB and E on AC. If AD ÷ AB = AE ÷ AC = k, then the dilation with centre A and factor k sends B to D and C to E, so △ADE is the image of △ABC. A dilation keeps angles and makes image sides parallel to the originals, so DE ∥ BC and DE = k × BC. This gives two tools. (1) If DE ∥ BC, then AD ÷ AB = AE ÷ AC = DE ÷ BC = k, and any unknown length follows. (2) To prove DE ∥ BC from lengths, show that AD ÷ AB = AE ÷ AC. If the two ratios differ, DE is not parallel to BC. When a triangle’s three sides are all multiplied by k, the new triangle is a dilation of the old one, and its perimeter is k times as big.

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Worked examples
In the picture AD = 6, DB = 4, AE = 9, EC = 6. Prove DE ∥ BC and find BC if DE = 12. AB = 6 + 4 = 10, AC = 9 + 6 = 15. AD ÷ AB = 6/10 = 3/5 and AE ÷ AC = 9/15 = 3/5. The ratios are equal, so △ADE is the image of △ABC under a dilation with centre A and k = 3/5. Hence DE ∥ BC. Also DE = 3/5 × BC, so BC = 12 ÷ 3/5 = 20.
Triangle ABC has sides 6, 8 and 10. Triangle A′B′C′ has sides 9, 12 and 15. Is it a dilation image, and what is k? 9 ÷ 6 = 12 ÷ 8 = 15 ÷ 10 = 3/2. All three sides are multiplied by 3/2, so k = 3/2. The perimeter grows from 24 to 36, also times 3/2.
Is DE ∥ BC if AD = 4, AB = 10, AE = 5, AC = 15? AD ÷ AB = 4/10 = 2/5 but AE ÷ AC = 5/15 = 1/3. The ratios differ, so no dilation from A sends B to D and C to E: DE is not parallel to BC.
Class activity

Parallel check. Draw triangle ABC on paper, mark D on AB and E on AC so that AD ÷ AB = AE ÷ AC. Draw DE. Using set squares (or by sliding a ruler), check that DE is parallel to BC. Then move E so that the ratios differ. What happens?

Practice
1
△ADE is the image of △ABC under a dilation with centre A. AD = 6, AB = 9, DE = 8. Find k and BC.
2
A triangle with sides 4, 6, 9 is dilated with k = 3/2. Find the new sides.
3
AD = 4, DB = 6, AE = 6, EC = 9. Is DE ∥ BC?
4
AD = 15, AB = 25, AE = 12, AC = 20, BC = 40. Is DE ∥ BC? Find DE.
5
The perimeter of △ABC is 30 and △ADE is its image with k = 2/5. Find the perimeter of △ADE.