SME
Grade 10 Math · Problem type
Altitude to the hypotenuse
Use this for “The altitude to the hypotenuse cuts it into 4 and 9. How long is the altitude?”
Diagram
This is an example. Type your own values to start a new problem.
h
2
=
p
×
q
c
Hypotenuse AB
calculated
b
Leg AC
calculated
a
Leg BC
calculated
Formulas
h
2
=
p
×
q
6
2
=
4
×
9
c
=
p
+
q
13 = 4 + 9
b
2
=
p
×
c
7.2111
2
=
4
×
13
a
2
=
q
×
c
10.8167
2
=
9
×
13
Assumptions
•
The altitude CD from the right angle splits △ABC into two triangles similar to it and to each other.
•
So the altitude is the geometric mean of the two parts: h² = pq.
•
Each leg is the geometric mean of the hypotenuse and the part next to it: b² = pc, a² = qc.
•
Lengths are in any one unit.
Step-by-step
WE KNOW
p = 4, q = 9
FIND
altitude CD (h), hypotenuse AB (c), leg AC (b), leg BC (a)
Step 1 · Find altitude CD (h)
Use h² = p × q
The altitude is the geometric mean of the two parts.
h
=
√
p
×
q
h
=
√
4 × 9
h
=
√
36
h
=
6
Step 2 · Find hypotenuse AB (c)
Use c = p + q
The two parts make the hypotenuse.
c
=
4
+
9
c
=
13
Step 3 · Find leg AC (b)
Use b² = p × c
Leg AC is the geometric mean of AD and AB.
b
=
√
p
×
c
b
=
√
4 × 13
b
=
√
52
b
=
7.2111
(2
√
13
)
Step 4 · Find leg BC (a)
Use a² = q × c
Leg BC is the geometric mean of DB and AB.
a
=
√
q
×
c
a
=
√
9 × 13
a
=
√
117
a
=
10.8167
(3
√
13
)
Check
Put the numbers back in. Both sides should match.
6
2
=
4
×
9
✓
13 = 4 + 9 ✓
7.2111
2
=
4
×
13
✓
10.8167
2
=
9
×
13
✓
Topic check
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Refresh
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