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Unit 8: Geometry and Trigonometry

Topic 4

The Pythagorean Theorem and Special Right Triangles

The Pythagorean Theorem states that in any right triangle with legs aa and bb and hypotenuse cc:
a2+b2=c2a^2 + b^2 = c^2
The hypotenuse is always the longest side and is always opposite the right angle. This theorem lets you find any one side of a right triangle when the other two are known. It also works in reverse: if three sides satisfy a2+b2=c2a^2 + b^2 = c^2, the triangle is a right triangle.
Certain sets of whole numbers satisfy the Pythagorean Theorem and appear frequently on the SAT. These are called Pythagorean triples. The most common are:
3-4-5,5-12-13,8-15-17,7-24-253\text{-}4\text{-}5, \quad 5\text{-}12\text{-}13, \quad 8\text{-}15\text{-}17, \quad 7\text{-}24\text{-}25
Any constant multiple of a Pythagorean triple is also a triple. For example, 3-4-53\text{-}4\text{-}5 gives 6-8-106\text{-}8\text{-}10, 9-12-159\text{-}12\text{-}15, 15-20-2515\text{-}20\text{-}25, and so on. Recognizing these triples saves time because you can identify the missing side by inspection instead of computing squares and square roots.
Two families of right triangles have side ratios worth memorizing because the SAT provides them on the reference sheet and tests them often:
A 45-45-90 triangle (isosceles right triangle) has sides in the ratio 1:1:21 : 1 : \sqrt{2}. If each leg has length ss, the hypotenuse is s2s\sqrt{2}. Conversely, if the hypotenuse is hh, each leg is h2=h22\dfrac{h}{\sqrt{2}} = \dfrac{h\sqrt{2}}{2}.
A 30-60-90 triangle has sides in the ratio 1:3:21 : \sqrt{3} : 2. The shortest side (opposite the 30∘30^\circ angle) has length ss, the side opposite the 60∘60^\circ angle has length s3s\sqrt{3}, and the hypotenuse (opposite the 90∘90^\circ angle) has length 2s2s. The most common mistake is mixing up which side gets the 3\sqrt{3} factor. Remember: the 3\sqrt{3} side is the middle-length side, between the shortest side and the hypotenuse.

Core Skills

  • Use the Pythagorean Theorem to find a missing side of a right triangle.
  • Recognize common Pythagorean triples and their multiples to save computation time.
  • Apply the side ratios of 45-45-90 and 30-60-90 triangles to find missing sides.
  • Use the converse of the Pythagorean Theorem to determine whether a triangle is a right triangle.
  • Solve multi-step problems that combine the Pythagorean Theorem with area, perimeter, or other geometry concepts.

Example 1: Basic Pythagorean Theorem

A right triangle has legs of length 9 and 12. What is the length of the hypotenuse? Step 1: Apply the Pythagorean Theorem.
c2=92+122=81+144=225c^2 = 9^2 + 12^2 = 81 + 144 = 225
Step 2: Take the square root.
c=225=15c = \sqrt{225} = 15
(You could also recognize this as the 3-4-53\text{-}4\text{-}5 triple scaled by 3: 9-12-159\text{-}12\text{-}15.) The hypotenuse is 15\boxed{15}.

Example 2: Finding a Leg

A right triangle has a hypotenuse of 26 and one leg of 10. What is the length of the other leg? Step 1: Let the unknown leg be aa.
a2+102=262  ⟹  a2+100=676a^2 + 10^2 = 26^2 \implies a^2 + 100 = 676
Step 2: Solve for aa.
a2=576  ⟹  a=24a^2 = 576 \implies a = 24
(This is the 5-12-135\text{-}12\text{-}13 triple scaled by 2: 10-24-2610\text{-}24\text{-}26.) The other leg is 24\boxed{24}.

Example 3: 45-45-90 Triangle

In a 45-45-90 triangle, the hypotenuse has length 10. What is the length of each leg? Step 1: In a 45-45-90 triangle, the hypotenuse is s2s\sqrt{2}, where ss is the leg length.
s2=10  ⟹  s=102=1022=52s\sqrt{2} = 10 \implies s = \frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}
Each leg has length 52\boxed{5\sqrt{2}}.

Example 4: 30-60-90 Triangle

In a 30-60-90 triangle, the side opposite the 30∘30^\circ angle has length 7. What are the lengths of the other two sides? Step 1: The shortest side (opposite 30∘30^\circ) is s=7s = 7. Step 2: The side opposite 60∘60^\circ is s3=73s\sqrt{3} = 7\sqrt{3}. Step 3: The hypotenuse is 2s=142s = 14. The side opposite 60∘60^\circ is 73\boxed{7\sqrt{3}} and the hypotenuse is 14\boxed{14}.

Example 5: Converse of the Pythagorean Theorem

A triangle has sides of length 11, 60, and 61. Is it a right triangle? Step 1: Check whether the square of the longest side equals the sum of the squares of the other two.
112+602=121+3600=372111^2 + 60^2 = 121 + 3600 = 3721
612=372161^2 = 3721
Since 112+602=61211^2 + 60^2 = 61^2, the triangle is a right triangle. Yes\boxed{\text{Yes}}

Example 6: Multi-Step Application

A rectangular room measures 8 feet by 15 feet. What is the length, in feet, of a diagonal of the room? Step 1: A diagonal of a rectangle divides it into two right triangles. The legs are the length and width of the room. Step 2: Apply the Pythagorean Theorem.
d2=82+152=64+225=289  ⟹  d=17d^2 = 8^2 + 15^2 = 64 + 225 = 289 \implies d = 17
(This is the 8-15-178\text{-}15\text{-}17 Pythagorean triple.) The diagonal is 17\boxed{17} feet.

Key Takeaways

  • Memorize the common Pythagorean triples (3-4-53\text{-}4\text{-}5, 5-12-135\text{-}12\text{-}13, 8-15-178\text{-}15\text{-}17) and watch for their multiples. Recognizing a triple instantly gives you the answer without any calculation.
  • In a 45-45-90 triangle, multiply a leg by 2\sqrt{2} to get the hypotenuse, or divide the hypotenuse by 2\sqrt{2} to get a leg. In a 30-60-90 triangle, the sides are ss, s3s\sqrt{3}, and 2s2s, with the 3\sqrt{3} always on the middle side (opposite 60∘60^\circ).
  • The most common error with 30-60-90 triangles is applying the 3\sqrt{3} to the wrong side. The hypotenuse is always exactly twice the shortest side, never involving 3\sqrt{3}.
  • When a problem gives you a rectangle, square, or equilateral triangle and asks for a diagonal or height, you are almost certainly using the Pythagorean Theorem or a special right triangle.