Similar Triangles and Scale Factors

What you’ll learn

Scale corresponding lengths, areas, and volumes appropriately.

ពង្រីកប្រវែង ផ្ទៃក្រឡា និងមាឌនៃរូបស្រដៀងតាមកត្តាត្រឹមត្រូវ។

Meaning and use

Similar triangles have equal corresponding angles and proportional corresponding sides. Identify matching vertices before taking ratios. A length scale factor k gives area factor k squared; for similar solids, volume factor k cubed. Area and volume do not usually scale by the same factor as length.

ត្រីកោណស្រដៀងមានមុំត្រូវគ្នាស្មើ និងជ្រុងត្រូវគ្នាសមាមាត្រ។ កត្តាប្រវែង k ធ្វើឱ្យផ្ទៃក្រឡាប្ដូរតាម k² និងមាឌរូបស្រដៀងតាម k³។

Formulas

Corresponding lengths

\[\frac{a'}a=\frac{b'}b=\frac{c'}c=k\]

Area and volume scale

\[A'=k^2A,\qquad V'=k^3V\]

Symbols and units

Symbol	Meaning	Unit or restriction
a, b, c and a′, b′, c′	Corresponding original and image side lengths	Common length unit
k	Positive length scale factor	Dimensionless
A, A′	Areas of similar shapes	Square unit
V, V′	Volumes of similar solids	Cubic unit

Conditions and limits

• Identify corresponding sides and establish similarity first.
• The volume rule requires similarity in all three dimensions.
• កុំយកជ្រុងដែលមិនត្រូវគ្នាមកបង្កើតផលធៀប។

Worked examples

A missing corresponding side

A 4 cm side corresponds to 10 cm; another original side is 6 cm.

1. Scale factor \(k=10/4=2.5\).
2. New side \(6\times2.5=15\) cm.

15 cm

រកកត្តាតាមជ្រុងត្រូវគ្នា រួចគុណជ្រុងផ្សេង។

Scaling an area

A similar shape has length scale factor 3. Original area is 12 cm².

1. Area factor \(3^2=9\).
2. New area \(12\times9=108\) cm².

108 cm²

កត្តាផ្ទៃក្រឡាជាការ៉េនៃកត្តាប្រវែង។

Common mistakes

Instead of	Use	Why
Doubling all lengths doubles the area.	Doubling all lengths multiplies area by 4.	Area scales with the square.
Any two triangles are similar.	Corresponding angles and side ratios must satisfy similarity.	Shape similarity is a condition to establish.

Practice

Solve these before revealing the worked solutions. Include units where relevant.

សាកល្បងដោះស្រាយមុនបើកមើលចម្លើយ។ សូមបង្ហាញជំហាន និងឯកតាដែលពាក់ព័ន្ធ។

1. A 5 cm side corresponds to 15 cm. Find k.
2. If k = 2 and area is 7 cm², find the new area.
3. For similar solids with k = 2 and volume 6 cm³, find the new volume.

Answers and steps

Show solutions / បង្ហាញដំណោះស្រាយ

1. 3

   15/5 = 3.

2. 28 cm²

   2² × 7 = 28.

3. 48 cm³

   2³ × 6 = 48.

Quick recap

• Match corresponding parts.
• Length, area, and volume use k, k², and k³.
• Establish similarity before applying the ratios.