SME
Grade 10 Math · Problem type
The side-splitter theorem
Use this for “DE ∥ BC, AD = 6, DB = 4 and AE = 9. Find EC.”
Diagram
DE ∥ BC, so it cuts the two sides in the same ratio.
AD
DB
=
6
4
=
3
2
AE
EC
=
9
6
=
3
2
Triangle ADE ~ triangle ABC, a dilation from A.
k
=
AD
AB
=
6
10
=
3
5
This is an example. Type your own values to start a new problem.
AD
DB
=
AE
EC
AB
Side AB
calculated
AC
Side AC
calculated
k
Scale factor AD ÷ AB
calculated
Formulas
AD
DB
=
AE
EC
6
4
=
9
6
AB = AD + DB
10 = 6 + 4
AC = AE + EC
15 = 9 + 6
AD
=
k
×
AB
6 = 0.6 × 10
AE
=
k
×
AC
9 = 0.6 × 15
Assumptions
•
D is on AB, E is on AC, and DE is parallel to BC.
•
A line parallel to one side cuts the other two sides in the same ratio: AD ÷ DB = AE ÷ EC.
•
△ADE is △ABC dilated from A by k = AD ÷ AB.
•
Lengths are in any one unit.
Step-by-step
WE KNOW
AD = 6, DB = 4, AE = 9
FIND
EC (EC), side AB (AB), side AC (AC), scale factor AD ÷ AB (k)
Step 1 · Find EC (EC)
Use AD ÷ DB = AE ÷ EC
A line parallel to one side cuts the other two in the same ratio. Cross-multiply, AD × EC = AE × DB, then divide by AD.
EC
=
AE
×
DB
AD
EC
=
9
×
4
6
EC
=
36
6
EC = 6
Step 2 · Find side AB (AB)
Use AB = AD + DB
Side AB is AD and DB put together.
AB = 6 + 4
AB = 10
Step 3 · Find side AC (AC)
Use AC = AE + EC
Side AC is AE and EC put together.
AC = 9 + 6
AC = 15
Step 4 · 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
=
6
10
k
=
0.6
Check
Put the numbers back in. Both sides should match.
6
4
=
9
6
✓
10 = 6 + 4 ✓
15 = 9 + 6 ✓
6 = 0.6 × 10 ✓
9 = 0.6 × 15 ✓
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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