Transcript Similar Triangles and Pythagorean Theorem

Similar Triangles and
Pythagorean Theorem
Section 6.4
Pythagorean Theorem
• Recall that a right triangle has a 90° angle as one of its
angles.
• The side that is opposite the 90° angle is called the
hypotenuse.
• The theorem due to Pythagoras says that the square of
the hypotenuse is equal to the sum of the squares of the
legs.
c2 = a2 + b2
a
c
b
Similar Triangles
•
•
Triangles are similar if two conditions are met:
1. The corresponding angle measures are equal.
2. Corresponding sides must be proportional. (That is, their
ratios must be equal.)
The triangles below are similar. They have the same shape,
but their size is different.
A
D
c
b
f
E
B
a
C
e
d
F
Corresponding angles and sides
• As you can see from the previous page we can
see that angle A is equal to angle D, angle B
equals angle E, and angle C equals angle F.
• The lengths of the sides are different but there is
a correspondence. Side a is in correspondence
with side d. Side b corresponds to side e. Side c
corresponds to side f.
• What we do have is a set of proportions.
• a/d = b/e = c/f
Example
• Find the missing side lengths for the
similar triangles.
3.2
3.8
y
54.4
x
42.5
ANSWER
• Notice that the 54.4 length side corresponds to
the 3.2 length side. This will form are complete
ratio.
• To find x, we notice side x corresponds to the
side of length 3.8.
• Thus we have 3.2/54.4 = 3.8/x. Solve for x.
• Thus x = (54.4)(3.8)/3.2 = 64.6
• Same thing for y we see that 3.2/54.4 = y/42.5.
Solving for y gives y = (42.5)(3.2)/54.4 = 2.5.