GED Mathematical Reasoning
Right Triangles & Pythagorean Theorem
A right triangle has one square corner. Its longest side, always opposite that square corner, is the hypotenuse. There is a fixed relationship between the three sides, and it is the single most useful formula on the math test — it turns any "how far diagonally" question into arithmetic.
The words first
- right angle
- a square corner, 90 degrees
- hypotenuse
- the longest side, always across from the right angle
- legs
- the two shorter sides that form the square corner
The method
THE FORMULA
a² + b² = c²
a and b are the two LEGS.
c is ALWAYS the hypotenuse — the long side across from the square corner.
FINDING THE LONG SIDE (you know both legs)
square them, add, then square-root
3² + 4² = 9 + 16 = 25 → c = √25 = 5
FINDING A SHORT SIDE (you know the long side and one leg)
square them, SUBTRACT, then square-root
c² − a² = b²
13² − 5² = 169 − 25 = 144 → b = √144 = 12
WORTH MEMORIZING: 3-4-5 and 5-12-13, and any multiple (6-8-10).
Worked all the way through
The problem
A ladder leans against a wall. Its base is 6 feet from the wall and it reaches 8 feet up. How long is the ladder?
- Step 1 — Draw it and find the square corner.The wall meets the ground at a right angle. The ground distance and the wall height are the two legs.
- Step 2 — Decide what the ladder is.The ladder is the slanted side across from the square corner, so it is the hypotenuse, c.
- Step 3 — Fill in the formula.6² + 8² = c²
- Step 4 — Square each number.36 + 64 = c²
- Step 5 — Add.100 = c²
- Step 6 — Undo the square with a square root.c = √100 = 10
On the board
a² + b² = c²
the formula
6² + 8² = c²
the legs go in for a and b
36 + 64 = c²
square each one
100 = c²
add
√100 = c
square root undoes the square
c = 10 ft
Answer
10 feet
A second one, in a different form
The hypotenuse of a right triangle is 13 and one leg is 5. Find the other leg.
a² + b² = c²
the formula
5² + b² = 13²
13 is the hypotenuse, so it is c
25 + b² = 169
b² = 169 − 25 = 144
SUBTRACT when finding a leg
b = √144 = 12
Answer
12
The usual mistake
Adding the legs directly. 6 and 8 give a hypotenuse of 10, not 14.
Check yourself
Check: the hypotenuse must be longer than either leg. 10 is longer than both 6 and 8. If your answer came out smaller than a leg, you subtracted when you should have added.
Practice it
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