SME
Grade 10 Math · Geometry
The Pythagorean theorem's converse and special right triangles
Use this for “A triangle has sides 8, 15 and 17. Is it a right, acute or obtuse triangle?”
Diagram
Three sides fix the triangle; the law of cosines gives each angle.
Law of sines: a/sin A = b/sin B = c/sin C ≈ 17
This is an example. Type your own values to start a new problem.
a
2
+
b
2
c
2
S
Sum of the squares, a² + b²
worked out (not typed)
Q
Square of c
worked out (not typed)
Formulas
S
=
a
2
+
b
2
289
=
8
2
+
15
2
Q
=
c
2
289
=
17
2
r
:
compare
S
with
Q
289 = 289, so right
Assumptions
•
c is the longest side. Compare a² + b² with c².
•
Equal: a right triangle (the converse of the Pythagorean theorem).
•
a² + b² less than c²: obtuse. More than c²: acute.
Step-by-step
WE KNOW
a = 8, b = 15, c = 17
FIND
sum of the squares, a² + b² (S), square of c (Q), kind of triangle (r)
Step 1 · Find sum of the squares, a² + b² (S)
Use S = a² + b²
Square the two shorter sides and add.
S
=
8
2
+
15
2
S
=
64
+
225
S
=
289
Step 2 · Find square of c (Q)
Use Q = c²
Square the longest side.
Q
=
17
2
Q
=
289
Step 3 · Find kind of triangle (r)
Use r: compare S with Q
a
2
+
b
2
equals
c
2
:
a
right
triangle,
the
right
angle
across
from
c
.
289 = 289: right
r
=
compare
289
with
289
r
:
=
Check
Put the numbers back in. Both sides should match.
289
=
8
2
+
15
2
✓
289
=
17
2
✓
289 = 289, so right ✓
Topic check
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Pythagorean theorem and distance between points
Exponent properties, rational exponents, simplifying radicals, and rational and irrational numbers