Apply the Fundamental Theorems of Proportionality to Solve.pptx

apply the fundamental theorem

1. Apply the Fundamental Theorems of Proportionality to Solve Problems Involving Proportion

2. Objective: 1. Applies the Fundamental T heorems of Proportionality to solve problems involving Proportions.

3. Definition of Similar Triangles Two Triangles are similar to each other if, The corresponding angles of both triangles are congruent, and The corresponding sides of both triangles are proportional. Thus, two triangles ABC and PQR are similar if, ABC AXY   B C A A X Y

4. A B C X Y In , if X and Y are points on and , respectively such that IS parallel to , then .  

5. Basic Proportional Theorem or BPT If a line is drawn parallel to one side of a triangle and intersects the other two sides in distinct points, then it divides the sides into segments which are proportional to these sides.   A B C X Y

6. EXAMPLE Given with . Formulate the possible proportions that can be derived given the triangle below following the basic proportionality theorem.   A B C M N a b c d k m

7.   A B C a 4 b 2 c 6 d 3 m 6 k 9

8. Let’s TRY!! To find the missing length in a triangle, we apply the Basic Proportionality Theorem and the Properties of Proportion. Consider where HL and EF are parallel to each other. EH = 9cm, HG = 21cm, FL= 6cm, Find LG. According to the Proportionality Theorem, Subbing in the known values leaves us with 9(LG) = 6 (21) 9LG = 126 LG = LG = 14cm   E F H L G

9. Try to solve! In the given at the right , Find . Set up Proportion. = = Solve for the missing length.   C D E B A 15mm 10mm

10. Try to solve! In the triangle below, , = 5cm. = x+6cm, = 3cm, and = x+3 cm. Find . = = Solve the equation.   B D C E A

11. Converse of the Basic Proportionality Theorem If a line intersects two sides of a triangle and the sides are divided into segments which are proportional, then the line is parallel to the third side. In , if X is a point between A and B and Y is a point between A and C and then .   A B C X Y

12. EXAMPLE 1 In , is ? Let us verify if the ratios of the sides are proportional. =   W 18 L 6 Y X 8 P 24

13. EXAMPLE 2 In and P are point on the sides , respectively. For each of the following , show that .   L M R O P