SME
Grade 10 Math · Problem type
The parallel side of a split triangle
Use this for “DE ∥ BC, AD = 3, AB = 12 and BC = 20. How long is DE?”
Diagram
DE ∥ BC, so it cuts the two sides in the same ratio.
AD
DB
=
3
9
=
1
3
AE
EC
=
4
12
=
1
3
Triangle ADE ~ triangle ABC, a dilation from A.
k
=
AD
AB
=
3
12
=
1
4
DE = k × BC = 1/4 × 20 = 5
This is an example. Type your own values to start a new problem.
DE
BC
=
AD
AB
AC
Side AC
example
DB
DB
calculated
k
Scale factor AD ÷ AB
calculated
AE
AE
calculated
Formulas
AB = AD + DB
12 = 3 + 9
AD
=
k
×
AB
3 = 0.25 × 12
AE
=
k
×
AC
4 = 0.25 × 16
DE
=
k
×
BC
5 = 0.25 × 20
Assumptions
•
DE is parallel to BC, with D on AB and E on AC.
•
△ADE is similar to △ABC with scale factor k = AD ÷ AB.
•
So DE = k × BC and AE = k × AC.
•
Lengths are in any one unit.
Step-by-step
WE KNOW
AB = 12, AC = 16, BC = 20, AD = 3
FIND
DB (DB), scale factor AD ÷ AB (k), AE (AE), DE (DE)
Step 1 · Find DB (DB)
Use AB = AD + DB
Side AB is AD and DB put together.
DB = AB − AD
DB = 12 − 3
DB = 9
Step 2 · Find scale factor AD ÷ AB (k)
Use AD = k × AB
The scale factor is a side of the image over its match in the original.
k
=
AD
AB
k
=
3
12
k
=
0.25
Step 3 · Find AE (AE)
Use AE = k × AC
Multiply the side AC by the scale factor.
AE = 0.25 × 16
AE = 4
Step 4 · Find DE (DE)
Use DE = k × BC
Multiply the side BC by the scale factor.
DE = 0.25 × 20
DE = 5
Check
Put the numbers back in. Both sides should match.
12 = 3 + 9 ✓
3 = 0.25 × 12 ✓
4 = 0.25 × 16 ✓
5 = 0.25 × 20 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Refresh
Translations, rotations, reflections and dilations
Proportional relationships and the constant of proportionality
Related lessons
Similarity: dilations and similar triangles
Main lesson
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