LESSON 5.4 Pythagorean Theorem OBJECTIVES: To use the Pythagorean theorem To solve real-life problems by using the Pythagorean theorem To use the converse of the Pythagorean theorem

The Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the legs.

Using the Pythagorean Theorem 6 8 x² 36 + 64 = x² 100= x²

Find the missing leg x 3 5² x²+ 9 = 25 x² = 16

Find the Area of the Triangle What is the formula for the area of a triangle? A = ½bh How will we find the height?

Find the Area of the Triangle

Converse of the Pythagorean Theorem If the square of the longest side is equal to the sum of the squares of the other sides then the triangle is a Right triangle.

Theorem If c2 is less then a2 + b2, then the triangle is Acute

Theorem If c2 is greater then a2 + b2, then the triangle is Obtuse

What type of Triangle Sides

What type of Triangle Sides

What type of Triangle Sides

What type of Triangle Sides

What type of Triangle Sides

What type of Triangle Sides

What type of Triangle Sides

What type of Triangle Sides

LESSON 5.6 Segments Divided Proportionally OBJECTIVES: To use proportionality theorems to calculate segment lengths To solve real-life problems by using proportions in triangles

THEOREM: TRIANGLE PROPORTIONALITY THEOREM If a line parallel to one side of a triangle, intersect the other two sides, then _____________ ______________ ______________. it divides the other two sides proportionally

THEOREM: CONVERSE OF TRIANGLE PROPORTIONALITY THEOREM If a line divides two sides of a triangle proportionally, then_________________ _____________________ _____________________ it is parallel to the third side.

EXAMPLE: Finding the length of a segment Use the Triangle Proportionality Theorem to find y. CM = ___ MB ___ = ___

EXAMPLE: Determining Parallel Lines Given the diagram, determine whether MN || GH. LM = ___ = ___ MG LN = ___ = ___ NH MN ___________ GH because________.

THEOREM: COROLLARY TO TRIANGLE PROPORTIONALITY THEOREM If three parallel lines intersect two transversals, then ______________ ______________ ______________. they divide the sides proportionally. a = __ b

EXAMPLE: Using the corollary to the Triangle Proportionality Theorem The segments joining the sides of trapezoid RSTU are parallel to its bases. Find x and y.

THEOREM: TRIANGLE-ANGLE- BISECTOR THEOREM If a ray bisects an angle of a triangle, then it ____________________ _____________________ _____________________ _____________________ _______. divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides. a = __; a = __ b p

EXAMPLE: Using the Triangle-Angle-Bisector Theorem Find the value of x. IG = ___ GH ___ = ___

AG = 6 GC = 4 CF = 4 FB = 3 AE = 5 Find BE

FINAL CHECKS FOR UNDERSTANDING 1. In the diagram, 1 2 3. PQ = 9, QR = 15, and ST = 11. What is the length of ?

Find x. 3. The real estate term for distance along the edge of a piece of property that touches the ocean is “ocean frontage.” Find the ocean frontage for each lot shown. Which of these lots should be listed for the highest price?

Mrs. McConaughy Geometry Homework Assignment: Mrs. McConaughy Geometry