Table of Contents
The Pythagorean Theorem
What is the Pythagorean theorem?
The area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.
Area1=Area2+Area3
AB2=BC2+AC2
C2=a2+b2
The square of the length of the hypotenuse is equal to the sum of the squares of the other two side lengths.
Under what conditions can you use the Pythagorean Theorem?
– In right angle triangles (one angle is 90°)
What is the hypotenuse?
-The longest side length
– Is always opposite to the right angle (90°)
What is the formula for the Area of a Triangle?
Example 1: In PQR, <Q is 90°, PR=25cm and QR=7cm, calculate the area.
Solution:
Area=bh2
=QRPQ2
-Use Pythagorean Theorem to find the height of the triangle, r=PQ.
q2=p2+r2PR2=QR2+PQ2
252=72+r2 sub. p and q
625=49+r2 solve for r
49+r2=625
r2=625-49
r2=576
r2=576take the square root of both sides
r=24
So, r=24 and r=-24 Inadmissible (There is no such thing. as a negative length)
∴ r=24cm
-Sub PQ=r=24 and QR=p=7 in
Area=pr2
=2472
=127
=84
∴ The area of PQR is 84cm2
Example 2: A field has dimensions 150m by 400m.
How much shorter is it to cut across the field diagonally then to walk around it?
Solution: -Draw a diagram
- Use P.Th. to find x
x2=4002+1502
x2=160000+22500
x2=182500
x2=182500
