SME
Grade 10 Math · Geometry
Similarity: dilations and similar triangles
Use this for “△ABC ∼ △DEF with sides 5, 6 and 8, and EF = 7.5. Find the scale factor and the other sides.”
Diagram
△DEF ∼ △ABC: the same angles, every side 1.5 times as long.
Side d = k × a = 1.5 × 5 = 7.5
Side e = k × b = 1.5 × 6 = 9
Side f = k × c = 1.5 × 8 = 12
This is an example. Type your own values to start a new problem.
d
a
=
e
b
=
f
c
=
k
Formulas
d
=
k
×
a
7.5 = 1.5 × 5
e
=
k
×
b
9 = 1.5 × 6
f
=
k
×
c
12 = 1.5 × 8
Assumptions
•
Similar triangles have equal angles, and every side of △DEF is k times its match in △ABC.
•
The scale factor k is a side of △DEF over the matching side of △ABC.
•
Match the sides in order: d with a, e with b, f with c.
•
Lengths are in any one unit.
Step-by-step
WE KNOW
a = 5, b = 6, c = 8, k = 1.5
FIND
side d (EF) (d), side e (FD) (e), side f (DE) (f)
Step 1 · Find side d (EF): d
Use d = k × a
Multiply the side by the scale factor.
d
=
1.5
×
5
d
=
7.5
Step 2 · Find side e (FD): e
Use e = k × b
Multiply the side by the scale factor.
e
=
1.5
×
6
e
=
9
Step 3 · Find side f (DE): f
Use f = k × c
Multiply the side by the scale factor.
f
=
1.5
×
8
f
=
12
Check
Put the numbers back in. Both sides should match.
7.5 = 1.5 × 5 ✓
9 = 1.5 × 6 ✓
12 = 1.5 × 8 ✓
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
More problem types
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?”
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?”
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?”