SME
Grade 8 Math · Problem type
Similar triangles
Use this for “Two triangles are similar. One has sides 8, 12 and x; the matching sides of the other are 6, y and 12. Find x and y.”
Diagram
b₁ = 12 on the top line sits over 9 on the bottom line: each 1 above is 3/4 below.
c₁ = 16 (first triangle, side c)
c₂ = 12 (second triangle, side c)
This is an example. Type your own values to start a new problem.
a₂
a₁
=
b₂
b₁
=
c₂
c₁
=
3
4
k
Formulas
k
=
a₂
÷
a₁
(scale
factor
=
second
triangle,
side
a
÷
first
triangle,
side
a)
3
4
=
6
÷
8
b₂
=
k
×
b₁
(second
triangle,
side
b
=
scale
factor
×
first
triangle,
side
b)
9
=
3
4
×
12
c₂
=
k
×
c₁
(second
triangle,
side
c
=
scale
factor
×
first
triangle,
side
c)
12
=
3
4
×
16
Assumptions
•
Similar triangles have the same angles; each side of the second is k times its match.
•
Find k from one pair of matching sides: second ÷ first.
•
Multiply by k to go from the first to the second; divide by k to go back.
Step-by-step
WE KNOW
a₁ = 8 (first triangle, side a), a₂ = 6 (second triangle, side a), b₁ = 12 (first triangle, side b), c₂ = 12 (second triangle, side c)
FIND
scale factor (k), second triangle, side b (b₂), first triangle, side c (c₁)
Step 1 · Find scale factor (k)
Use k = a₂ ÷ a₁ (scale factor = second triangle, side a ÷ first triangle, side a)
The second side divided by its match.
k
=
6
8
k
=
3
4
Step 2 · Find second triangle, side b (b₂)
Use b₂ = k × b₁ (second triangle, side b = scale factor × first triangle, side b)
Multiply
the
first
triangle’s
side
by
k
.
b₂
=
3
4
×
12
b₂
=
9
Step 3 · Find first triangle, side c (c₁)
Use c₂ = k × c₁ (second triangle, side c = scale factor × first triangle, side c)
Go
back:
divide
the
second
triangle’s
side
by
k
.
12
=
3
4
×
c₁
c₁
=
12
3
4
c₁
=
16
Check
Put the numbers back in. Both sides should match.
3
4
=
6
÷
8
✓
9
=
3
4
×
12
✓
12
=
3
4
×
16
✓
Topic check
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