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Wize High School Grade 10 Math Textbook > Trigonometry of Right Triangles

Pythagorean Therom (Review)

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Pythagorean Theorem

The side lengths of a right angle triangle have a very special relationship.
   c2  =  a2  +  b2   \Large\boxed{~~~\bcf c^2~~=~~\bcth a^2~~+~~\bct b^2~~~}


Wize Tip
  • c\bcf c is the hypotenuse, which is the longest side in a right angle triangle across from the right angle.
  • It does NOT matter which of the other 2 sides you use as a\bcth a or b\bct b.

We can rearrange this formula to solve for any of the sides:
  • c=a2+b2\bcf c=\sqrt{\bcth a^2+\bct b^2}
  • a=c2−b2\bcth a=\sqrt{\bcf c^2-\bct b^2}
  • b=c2−a2\bct b=\sqrt{\bcf c^2-\bcth a^2}
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Example: Pythagorean Theorem

Determine the missing lengths in each of the following triangles.

a)

The missing side is the hypotenuse:
  • c=?\bcf {c=?}
  • a=3 cm\bcth {a=3~cm}
  • b=4 cm\bct{b=4~cm}
Using the Pythagorean Theorem:
c2=a2+b2\bcf c^2=\bcth a^2+\bct b^2
c=a2+b2\bcf c=\sqrt{\bcth a^2+\bct b^2}
c=32+42\bcf c=\sqrt{\bcth 3^2+\bct 4^2}
c=9+16\bcf c=\sqrt{\bcth 9+\bct {16}}
c =25\bcf c~=\sqrt{25}
c =5 cm\boxed{\bcf c~=5~cm}

This is a special 3-4-5 triangle.

PAGE BREAK
b)


The hypotenuse has length 11m:
  • c=11 m\bcf {c=11~m}
  • a=?\bcth {a=?}
  • b=7 cm\bct{b=7~cm}
Using the Pythagorean Theorem:
c2=a2+b2\bcf c^2=\bcth a^2+\bct b^2
a=c2−b2\bcth a=\sqrt{\bcf c^2-\bct b^2}
a=112−72\bcth a=\sqrt{\bcf {11}^2-\bct 7^2}
a=121−49\bcth a=\sqrt{\bcf {121}-\bct {49}}
a=72\bcth a=\sqrt{72}
a≈8.485 m\boxed{\bcth a\approx 8.485~m}

Practice: Pythagorean Theorem

Find the missing side lengths in the following triangles.

a.

b.

Practice: Pythagorean Theorem

Calculate the missing side lengths.

a)


b)

Practice: Pythagorean Theorem

For any cones or pyramids,
  • the height is the length that is perpendicular to the base of the figure.
  • the slant height is the length of the diagonal that goes from the vertex (tip) of the figure to the edge of the base.

The diameter of the base of a cone is 10cm and the slant height is 8cm, determine the height of this cone.