Similar Triangles and Pythagorean Theorem Section 6.4.

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Presentation transcript:

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.c 2 = a 2 + b 2 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 E d F B a C

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 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.