Wize High School Grade 10 Math Textbook > Analytic Geometry

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~~~}   c2  =  a2  +  b2   ​


Wize Tip
c\bcf cc 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 aa or b\bct bb.

We can rearrange this formula to solve for any of the sides:

c=a2+b2\bcf c=\sqrt{\bcth a^2+\bct b^2}c=a2+b2​
a=c2−b2\bcth a=\sqrt{\bcf c^2-\bct b^2}a=c2−b2​
b=c2−a2\bct b=\sqrt{\bcf c^2-\bcth a^2}b=c2−a2​

0:00 / 0:00

Example: Pythagorean Theorem
Determine the missing lengths in each of the following triangles.

a) 

The missing side is the hypotenuse:
c=?\bcf {c=?}c=?
a=3 cm\bcth {a=3~cm}a=3 cm
b=4 cm\bct{b=4~cm}b=4 cm
Using the Pythagorean Theorem:
c2=a2+b2\bcf c^2=\bcth a^2+\bct b^2c2=a2+b2
c=a2+b2\bcf c=\sqrt{\bcth a^2+\bct b^2}c=a2+b2​
c=32+42\bcf c=\sqrt{\bcth 3^2+\bct 4^2}c=32+42​
c=9+16\bcf c=\sqrt{\bcth 9+\bct {16}}c=9+16​
c =25\bcf c~=\sqrt{25}c =25​
c =5 cm\boxed{\bcf c~=5~cm}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}c=11 m
a=?\bcth {a=?}a=?
b=7 cm\bct{b=7~cm}b=7 cm
Using the Pythagorean Theorem:
c2=a2+b2\bcf c^2=\bcth a^2+\bct b^2c2=a2+b2
a=c2−b2\bcth a=\sqrt{\bcf c^2-\bct b^2}a=c2−b2​
a=112−72\bcth a=\sqrt{\bcf {11}^2-\bct 7^2}a=112−72​
a=121−49\bcth a=\sqrt{\bcf {121}-\bct {49}}a=121−49​
a=72\bcth a=\sqrt{72}a=72​
a≈8.485 m\boxed{\bcth a\approx 8.485~m}a≈8.485 m​

Practice: Pythagorean Theorem
Calculate the missing side lengths.

a) 

b)

a) Enter the missing side length rounded to 2 decimal places, do not include units.

b) Enter the missing side length rounded to 2 decimal places, do not include units.

I don't know