SME
Grade 10 Math · Problem type
Is DE parallel to BC?
Use this for “D is on AB and E is on AC; AD = 4, DB = 6, AE = 6 and EC = 9. Is DE parallel to BC?”
Diagram
This is an example. Type your own values to start a new problem.
AD
DB
AE
EC
u
Ratio AD ÷ DB
worked out (not typed)
w
Ratio AE ÷ EC
worked out (not typed)
Formulas
u
=
AD
DB
2
3
=
4
6
w
=
AE
EC
2
3
=
6
9
g
:
compare
u
with
w
2
3
=
2
3
,
so
parallel
Assumptions
•
D is on side AB and E is on side AC.
•
Converse of the side-splitter theorem: DE ∥ BC exactly when AD ÷ DB = AE ÷ EC.
•
Unequal ratios mean DE tilts toward BC and would meet it.
Step-by-step
WE KNOW
AD = 4, DB = 6, AE = 6, EC = 9
FIND
ratio AD ÷ DB (u), ratio AE ÷ EC (w), whether DE ∥ BC (g)
Step 1 · Find ratio AD ÷ DB (u)
Use u = AD ÷ DB
How many times DB fits in AD.
u
=
4
6
u
=
2
3
Step 2 · Find ratio AE ÷ EC (w)
Use w = AE ÷ EC
How many times EC fits in AE.
w
=
6
9
w
=
2
3
Step 3 · Find whether DE ∥ BC (g)
Use g: compare u with w
The ratios are equal, so DE is parallel to BC: the converse of the side-splitter theorem.
2
3
=
2
3
:
parallel
g
=
compare
2
3
with
2
3
g
:
=
Check
Put the numbers back in. Both sides should match.
2
3
=
4
6
✓
2
3
=
6
9
✓
2
3
=
2
3
,
so
parallel
✓
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
Dilation from a center
Use this for “Dilate an 8 by 6 rectangle by a scale factor of 0.5 from the point O.”
The side-splitter theorem
Use this for “DE ∥ BC, AD = 6, DB = 4 and AE = 9. Find EC.”
The parallel side of a split triangle
Use this for “DE ∥ BC, AD = 3, AB = 12 and BC = 20. How long is DE?”
Scale factor, perimeter and area
Use this for “A 3 by 4 rectangle is enlarged by 2.5. What are the new perimeter and area?”
Altitude to the hypotenuse
Use this for “The altitude to the hypotenuse cuts it into 4 and 9. How long is the altitude?”
Similar, and by which test?
Use this for “Which pairs of triangles must be similar, and why?”