📐 Geometry · Similarity and Right Triangles
The Pythagorean Theorem flashcards
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All 17 flashcards at a glance
Card 1
State the Pythagorean theorem.
In a right triangle with legs a and b and hypotenuse c: a2 + b2 = c2.
Card 2
What is the hypotenuse, and how do you find it in a picture?
The side across from the right angle. It does not touch the right angle, and it is the longest side.
Card 3
What are the legs of a right triangle?
The two sides that meet at the right angle.
Card 4
Find the hypotenuse of a right triangle with legs 5 and 9.
52 + 92 = 25 + 81 = 106, so c = √106 ≈ 10.30. (106 has no perfect-square factor.)
Card 5
Find the other leg of a right triangle with hypotenuse 10 and one leg 4.
b2 = 102 − 42 = 100 − 16 = 84, so b = √84 = 2√21 ≈ 9.17.
Card 6
Simplify √50.
√50 = √25 × √2 = 5√2 ≈ 7.07
Card 7
Simplify √48.
√48 = √16 × √3 = 4√3 ≈ 6.93
Card 8
What is the converse of the Pythagorean theorem?
If the side lengths of a triangle satisfy a2 + b2 = c2, where c is the longest side, then the triangle is a right triangle.
Card 9
A triangle has sides 6, 9 and 7. Which side plays the role of c, and is it a right triangle?
c = 9, the longest side. 62 + 72 = 36 + 49 = 85, but 92 = 81, so it is not a right triangle.
Card 10
Name the three most common Pythagorean triples.
3-4-5, 5-12-13 and 8-15-17. Check: 9 + 16 = 25, 25 + 144 = 169, 64 + 225 = 289.
Card 11
How do you make new Pythagorean triples from a known one?
Multiply all three numbers by the same positive whole number. For example, 3-4-5 times 10 is 30-40-50, since 900 + 1600 = 2500.
Card 12
Why is c not equal to a + b in a right triangle?
Going along both legs is farther than going straight across. For legs 3 and 4, a + b = 7 but c = 5. The correct relationship is c2 = a2 + b2.
Card 13
A student writes √13 = 3.61. What is wrong?
3.61 is a rounded value, so write √13 ≈ 3.61. The exact value is √13, and its decimal digits never end.
Card 14
Why is the hypotenuse longer than each leg?
Because c2 = a2 + b2 is bigger than a2 alone, so c > a. In the same way c > b.
Card 15
In a rectangle, what is the diagonal in terms of a right triangle?
The hypotenuse. The rectangle’s width and height are the legs.
Card 16
How can you tell that a hypotenuse answer is reasonable?
It must be longer than each leg and shorter than the sum of the two legs. A missing leg must be shorter than the hypotenuse.
Card 17
What is the formula for a missing leg b, given hypotenuse c and the other leg a?
b = √(c2 − a2). Subtract the known leg’s square from the hypotenuse’s square. Never add.