LESSON 5A.3 P R OV I N G P Y T H AG O R E A N

A P P L I C AT I O N S O F S I M I L A R T R I A N G L E S

Section 5.4.3. & 5.4.4

LESSON 5A.3 P R OV I N G P Y T H AG O R E A N

A P P L I C AT I O N S O F S I M I L A R T R I A N G L E S

Section 5.4.3. & 5.4.4

PYTHAGOREAN THEOREM

�Once a statement has been shown to be true, it is called a theorem.

�One of the most well known theorems of geometry is the Pythagorean Theorem, which relates the length of the hypotenuse of a right triangle to the lengths of its legs.

�The Pythagorean Theorem is often used to find the lengths of the sides of a right triangle, a triangle that includes one 90° angle.

Section 5.4.3

PYTHAGOREAN THEOREM

Section 5.4.3

CONVERSE OF PYTHAGOREAN THEOREM

• It is also true that if the sum of the squares of the measures of two sides of a triangle equals the square of the measure of the longest side, then the triangle is a right triangle.

• This is known as the converse of the Pythagorean Theorem.

Section 5.4.3

PROOFS • Paragraph proofs are statements written out in complete

sentences in a logical order to show an argument.

• Flow proofs are a graphical method of presenting the logical steps used to show an argument.

• In a flow proof, the logical statements are written in boxes and the reason for each statement is written below the box.

• Another accepted form of proof is a two-column proof.

• Two-column proofs include numbered statements and corresponding reasons that show the argument in a logical order.

Section 5.4.3

EXAMPLE 1 (2 COLUMN PROOF)

Let's just look at example one. This is a 2 column proof looks like.

Section 5.4.3

EXAMPLE 2

Find the length of the altitude, x, of .

Section 5.4.3

EXAMPLE 3 Find the unknown values in the figure.

Section 5.4.3

TASK 5.4.3 (PG. U5-182) A cell phone company is interested in placing a 5-meter-tall antenna on an existing tower in order to boost their cell signal without having to build a new tower. A surveyor standing 7.5 meters from the base of the tower calculates the height of the existing tower. The line of sight from the lens on the surveyor’s tripod to the base of the tower is 1 meter above the ground. If the 5-meter-tall antenna is added to the top of the tower, will the company be required to add lighting to the existing structure? Explain your reasoning.

Section 5.4.3

USING SIMILAR TRIANGLES Previously we learned how to prove similar triangles, now we are going to use those methods to solve story problems and apply them to real life.

Recall:

• Similarity statements include Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS). Similar triangles have corresponding sides that are proportional.

• The Triangle Proportionality Theorem states that if a line parallel to one side of a triangle intersects the other two sides of the triangle, then the parallel line divides these two sides proportionally.

• The Triangle Angle Bisector Theorem states if one angle of a triangle is bisected, or cut in half, then the angle bisector of the triangle divides the opposite side of the triangle into two segments that are proportional to the other two sides of the triangle.

Section 5.4.4

BEWARE Common Errors/Misconceptions • misidentifying congruent angles because of the

orientation of the triangles

• incorrectly creating proportions between corresponding sides

• assuming a line parallel to one side of a triangle bisects the remaining sides rather than creating proportional sides

• misidentifying the altitudes of triangles

• incorrectly simplifying expressions with square roots

Section 5.4.4

EXAMPLE 1 A meter stick casts a shadow 65 centimeters long. At the same time, a tree casts a shadow 2.6 meters long. How tall is the tree?

Section 5.4.4

EXAMPLE 2

Section 5.4.4

EXAMPLE 3

Section 5.4.4

EXAMPLE 4 To estimate the height of an overhang, a surveyor positions herself so that her line of sight to the top of the overhang and her line of sight to the bottom form a right angle. What is the height of the overhang to the nearest tenth of a meter?

Section 5.4.4

ASSIGNMENT 5A.3

�WB Pg U5-186 #’s 1-4

�WB Pg U5-209 #’s 1-10