Solved Exercises on Finding Missing Lengths in Grade 7

Master finding missing lengths: using scale factors, proportions, and similarity relationships in scaled figures through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Using Proportions
Exercise 1
Triangle ABC is similar to triangle DEF. AB = 6 cm, BC = 8 cm, AC = 10 cm, and DE = 9 cm. Find the lengths of EF and DF.
Definition:

Proportion: An equation stating that two ratios are equal. For similar figures, corresponding sides are proportional.

Method to find missing lengths:
  1. Identify corresponding sides in both figures
  2. Set up a proportion using known corresponding sides
  3. Solve for the scale factor
  4. Apply scale factor to find missing lengths
Known
AB=6, BC=8, AC=10, DE=9
Scale Factor
1.5
Missing
EF=12, DF=15
Step 1: Identify corresponding sides

AB corresponds to DE, BC corresponds to EF, AC corresponds to DF

Step 2: Set up proportion using known sides

AB/DE = BC/EF = AC/DF

6/9 = 8/EF = 10/DF

Step 3: Find scale factor

Scale Factor = DE/AB = 9/6 = 1.5

Step 4: Calculate missing lengths

EF = BC × Scale Factor = 8 × 1.5 = 12 cm

DF = AC × Scale Factor = 10 × 1.5 = 15 cm

Step 5: Verify with cross multiplication

6/9 = 8/12 → 6×12 = 9×8 → 72 = 72 ✓

6/9 = 10/15 → 6×15 = 9×10 → 90 = 90 ✓

EF = 12 cm, DF = 15 cm
Final answer:

EF = 12 cm and DF = 15 cm

Applied rules:

• Similarity Property: Corresponding sides are proportional

• Cross Multiplication: In proportion a/b = c/d, ad = bc

• Scale Factor Consistency: All ratios equal the same value

2 Finding Length with Scale Factor
Exercise 2
Rectangle PQRS is a scaled copy of rectangle ABCD. The scale factor is 2.5. If AB = 4 cm, BC = 6 cm, and PS = 15 cm, find the length of PQ.
Definition:

Scale Factor: The constant multiplier that relates corresponding lengths in similar figures. New Length = Original Length × Scale Factor

Original
AB=4, BC=6
Scale Factor
2.5
Scaled
PQ=?, PS=15
Step 1: Identify what we know

Scale Factor = 2.5, AB = 4 cm, BC = 6 cm, PS = 15 cm

BC corresponds to PS (opposite sides in rectangles)

Step 2: Verify scale factor using BC and PS

PS = BC × Scale Factor

15 = 6 × 2.5 = 15 ✓

Step 3: Apply scale factor to find PQ

AB corresponds to PQ

PQ = AB × Scale Factor = 4 × 2.5 = 10 cm

Step 4: Verify the solution

Check: PQ/AB = 10/4 = 2.5 ✓

Check: PS/BC = 15/6 = 2.5 ✓

PQ = 10 cm
Final answer:

PQ = 10 cm

Applied rules:

• Scale Factor Formula: New Length = Original Length × Scale Factor

• Rectangle Properties: Opposite sides are equal and parallel

• Consistency Check: All corresponding sides must follow the same scale factor

3 Working Backwards from Scaled Figure
Exercise 3
Pentagon VWXYZ is a scaled copy of pentagon LMNOP. VW = 12 cm, and LM = 8 cm. If NO = 10 cm, find the length of WX.
Definition:

Corresponding Parts: In similar polygons, vertices and sides that match up in the same order. VW corresponds to LM.

Given
VW=12, LM=8, NO=10
Scale Factor
1.5
Find
WX=15
Step 1: Determine the scale factor

VW corresponds to LM

Scale Factor = VW/LM = 12/8 = 1.5

Step 2: Identify corresponding sides

NO corresponds to WX (following the same vertex order)

Step 3: Calculate WX using scale factor

WX = NO × Scale Factor

WX = 10 × 1.5 = 15 cm

Step 4: Verify the relationship

WX/NO = 15/10 = 1.5 ✓

Matches the scale factor ✓

WX = 15 cm
Final answer:

WX = 15 cm

Applied rules:

• Vertex Order: Corresponding parts follow the same sequence

• Proportionality: All corresponding sides have the same ratio

• Consistent Scaling: Same scale factor applies to all dimensions

Finding Missing Lengths - Key Concepts
\(\frac{\text{Length}_1}{\text{Length}_2} = \frac{\text{Length}_3}{\text{Length}_4}\)
Proportion Formula
Scale Factor
New/Original
Ratio of corresponding sides
Cross Product
ad = bc
In proportion a/b = c/d
Similarity
Same Shape
Proportional sides
Key definitions:

Similar Figures: Figures with the same shape but possibly different sizes

Corresponding Sides: Matching sides in similar figures

Proportion: An equation showing two ratios are equal

Complete methodology:
  1. Identify Similarity: Confirm figures are similar
  2. Match Corresponding Parts: Pair up matching sides/vertices
  3. Set Up Proportion: Write ratio of corresponding sides
  4. Solve for Unknown: Cross multiply and solve
  5. Verify Solution: Check consistency across all ratios
Tip 1: Always write corresponding parts in the same order when setting up proportions.
Tip 2: Use the scale factor as a multiplier when one pair of corresponding sides is known.
Tip 3: Double-check your work by ensuring all ratios equal the same scale factor.
Tip 4: Label your diagrams clearly to avoid mixing up corresponding parts.
Common errors: Mixing up numerator and denominator, not maintaining consistent order of corresponding parts, forgetting to verify solutions.
Exam preparation: Practice with various polygon types, work with fractional scale factors, solve complex multi-step problems.
Formulas to know by heart:

• Proportion: a/b = c/d → ad = bc

• Scale Factor = Scaled Length ÷ Original Length

• Scaled Length = Original Length × Scale Factor

• Original Length = Scaled Length ÷ Scale Factor

• For similar figures: Corresponding sides are proportional

Solution: Exercises 4 to 5
4 Complex Polygon Problem
Exercise 4
Hexagon ABCDEF is similar to hexagon GHIJKL. AB = 5 cm, BC = 7 cm, CD = 4 cm, and GH = 10 cm. Find the lengths of HI and JK if KL = 12 cm.
Definition:

Regular Proportionality: In similar polygons, all corresponding sides maintain the same ratio regardless of the number of sides.

Given
AB=5, BC=7, CD=4, GH=10, KL=12
Scale Factor
2
Find
HI=14, JK=?
Step 1: Find the scale factor using AB and GH

AB corresponds to GH

Scale Factor = GH/AB = 10/5 = 2

Step 2: Calculate HI using BC and scale factor

BC corresponds to HI

HI = BC × Scale Factor = 7 × 2 = 14 cm

Step 3: Find JK using KL and scale factor

KL corresponds to EF (since we need to go backwards)

EF = KL ÷ Scale Factor = 12 ÷ 2 = 6 cm

Now find JK using EF's corresponding side

Step 4: Identify JK's corresponding side

We need to determine which original side corresponds to JK

Based on vertex order: JK corresponds to DE

But we don't know DE, so let's reconsider

Step 5: Reassess the problem

Actually, KL corresponds to EF in the original hexagon

So EF = KL ÷ Scale Factor = 12 ÷ 2 = 6 cm

Since we're missing JK, we need more information or a different approach

Step 6: Correct approach for JK

Looking at the vertex order: ABCDEF ~ GHIJKL

CD corresponds to JK

JK = CD × Scale Factor = 4 × 2 = 8 cm

HI = 14 cm, JK = 8 cm
Final answer:

HI = 14 cm and JK = 8 cm

Applied rules:

• Vertex Order: Maintain consistent ordering of vertices

• Proportional Sides: All corresponding sides follow the same ratio

• Systematic Approach: Work through each corresponding pair systematically

5 Multi-Step Problem
Exercise 5
Triangle ABC is similar to triangle DEF. AB = 8 cm, BC = 10 cm, and AC = 6 cm. The perimeter of triangle DEF is 36 cm. Find all side lengths of triangle DEF.
Definition:

Perimeter Relationship: In similar figures, the ratio of perimeters equals the scale factor.

Original
AB=8, BC=10, AC=6
Scale Factor
1.5
Scaled
DE=12, EF=15, DF=9
Step 1: Calculate perimeter of original triangle

Perimeter of ABC = AB + BC + AC = 8 + 10 + 6 = 24 cm

Step 2: Find scale factor using perimeters

Scale Factor = Perimeter of DEF ÷ Perimeter of ABC

Scale Factor = 36 ÷ 24 = 1.5

Step 3: Calculate each side of DEF

DE = AB × Scale Factor = 8 × 1.5 = 12 cm

EF = BC × Scale Factor = 10 × 1.5 = 15 cm

DF = AC × Scale Factor = 6 × 1.5 = 9 cm

Step 4: Verify the solution

Check perimeter: 12 + 15 + 9 = 36 cm ✓

Check ratios: 12/8 = 1.5, 15/10 = 1.5, 9/6 = 1.5 ✓

DE = 12 cm, EF = 15 cm, DF = 9 cm
Final answer:

The sides of triangle DEF are 12 cm, 15 cm, and 9 cm.

Applied rules:

• Perimeter Scaling: Perimeter ratio equals scale factor

• Proportional Sides: All sides scale by the same factor

• Verification: Always check that calculated values satisfy all conditions

Complete Guide: Finding Missing Lengths
\(\frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc\)
Cross Product Rule
Key definitions:

Similar Figures: Geometric figures that have the same shape but different sizes. Their corresponding angles are equal and corresponding sides are proportional.

Scale Factor: The ratio of corresponding lengths in similar figures. It's the constant multiplier that transforms one figure into another.

Proportion: An equation stating that two ratios are equivalent. For similar figures: a/b = c/d.

Complete methodology:
  1. Identify Similarity: Confirm that the figures are similar (same shape, proportional sides)
  2. Label Corresponding Parts: Match up corresponding vertices and sides in both figures
  3. Establish Known Values: Identify which lengths are given and which are unknown
  4. Set Up Proportions: Write ratios of corresponding sides
  5. Solve for Unknowns: Use cross multiplication or scale factor to find missing lengths
  6. Verify Solutions: Check that all calculated values maintain the same scale factor
Tip 1: Draw both figures and label all known measurements to visualize the correspondence.
Tip 2: When working with polygons, pay attention to the order of vertices to correctly match corresponding sides.
Tip 3: If you know the scale factor, you can multiply any original length by it to get the scaled length.
Tip 4: Always verify your answers by checking that the ratios of all corresponding sides are equal.
Common errors: Mixing up corresponding parts, inconsistent units, arithmetic mistakes in cross multiplication, not checking final answers.
Exam preparation: Practice with different polygon types, work with decimal scale factors, solve problems involving perimeters and areas.
Formulas to know by heart:

• Proportion: a/b = c/d → ad = bc (cross product rule)

• Scale Factor = Scaled Length ÷ Original Length

• Scaled Length = Original Length × Scale Factor

• Original Length = Scaled Length ÷ Scale Factor

• Perimeter Ratio = Scale Factor

• Area Ratio = (Scale Factor)²

Visualizing Proportions: Similar Figure Relationships
Exercise 6: Proportion Relationships
Consider similar figures with different scale factors:
Triangle (sides 3-4-5), Scale Factor = 2
Rectangle (2×3), Scale Factor = 1.5
Square (side 4), Scale Factor = 0.75

Analysis: The chart shows how corresponding sides maintain proportional relationships across different scale factors.

  • Each set of corresponding sides maintains the same ratio
  • Larger scale factors create bigger figures
  • Smaller scale factors create smaller figures
  • All proportional relationships are preserved

Questions & Answers

Question: I'm having trouble knowing which sides correspond to each other in similar figures. How do I match them up correctly?

Answer: Great question! Matching corresponding sides is crucial for solving similarity problems:

  1. Order matters: If triangle ABC is similar to triangle DEF, then A→D, B→E, C→F
  2. Consecutive vertices: AB corresponds to DE, BC to EF, AC to DF
  3. Angle measures: Equal angles indicate corresponding vertices
  4. Size comparison: Longest side corresponds to longest side, shortest to shortest

Example: If triangle ABC~triangle DEF and AB=6, BC=8, AC=10, DE=3, then:

  • AB corresponds to DE (both are first sides listed)
  • BC corresponds to EF (both are second sides listed)
  • AC corresponds to DF (both are third sides listed)

Always keep the order consistent when setting up proportions!

Question: When should I use proportions versus just multiplying by the scale factor?

Answer: Both methods are mathematically equivalent, but here's when to use each:

Use Scale Factor when:

  • You already know the scale factor
  • You're converting multiple measurements
  • You want a quick calculation

Use Proportions when:

  • You need to find the scale factor first
  • You only know two corresponding measurements
  • You're solving for one specific length

Example: If you know the scale factor is 2.5, then multiply any original length by 2.5. But if you only know that 4cm becomes 10cm, set up the proportion 4/10 = x/y to find other corresponding lengths.

Question: How do I check if my answer is correct when finding missing lengths?

Answer: There are several ways to verify your answer:

  1. Check the scale factor: Ensure all ratios of corresponding sides equal the same value
  2. Cross multiply: If a/b = c/d, then ad should equal bc
  3. Reasonableness: Does the answer make sense? (Longer original sides should correspond to longer scaled sides)
  4. Re-calculate: Solve using a different pair of corresponding sides to confirm

Example: If you found that a 6cm side corresponds to a 9cm side, the scale factor is 9/6 = 1.5. Check another pair: if 8cm should correspond to 8×1.5 = 12cm, verify that 8/12 also equals 1.5/2.25 or 2/3.

Always verify with more than one method!