MODULE 8

• Discipline

  Good Taste
• Writers: John Anthony P. Santos, Chonalyn L. Tecson
• Cover Illustrator: Joel J. Estudillo
• This module is for Mathematics, Quarter 3: Module 8
• The module is about Applying Similarity Theorems and Proving Pythagorean Theorem
• This module is from the Department of Education, National Capital Region, Schools Division Office, Marikina City
• What I Need to Know


  1. Apply the theorems to show that the given triangles are similar (M9GE-IIIi-1), and b) Prove the Pythagorean Theorem. (M9GE-IIIi-1)
• What I Know

  After going through this module, you are expected to: 1. verify the similarity theorems, 2. apply the properties of similar triangles and the similarity and proportionality theorems to calculate lengths of a certain line segments, 3. apply the theorem to show that the given triangles are similar, and 4. prove the Pythagorean theorem.
• The module is divided into two lessons: Lesson 1 - Applying the theorems to show that the given triangles are similar, Lesson 2 - Proving Pythagorean Theorem
• Multiple choice questions


  • ∆ABC~∆DEF. If DE = 2cm, AB and BC = 3cm, then EF is equal to _______.
  • Given: CI bisects ∠OCN and ∠OIN. Which theorem can be applied to show that ∆OCN ~ ∆OIN?
  • What value of x would make the triangles similar?
  • ∆ABC ~ ∆DEF. Find the perimeter of ∆DEF.
  • If the shadow of a tree is 14cm long and the shadow of the person who is 1.8m tall is 4m long, how tall is the tree?
  • If side AR = 3 and side AC = 4. How long is side CR?
  • If side AC = 12 and side CR = 15. How long is side AR?
  • If side AR = 30 and side AC = 40, then side CR = 50.
  • What is the length of side AC if AR = 6cm and CR = 10cm?
  • 10. What is the length of side AC if AR = 5cm and CR = 13cm?
• Lesson 1: Applying the Theorems to Show that the Given Triangles are Similar
• Similar Triangles


  Two triangles are similar if corresponding angles are congruent and corresponding sides are proportional.
• Are the triangles similar or not?
• AAA (Angle-Angle-Angle) Similarity Postulate


  If the corresponding angles of the two triangles are congruent, then the two triangles are similar.
• AA (Angle-Angle) Similarity Theorem


  If two angles of one triangle are congruent to their corresponding angles in of another triangle, then the two triangles are similar.
• SSS (Side-Side-Side) Similarity Theorem


  If the sides of one triangle are proportional to the corresponding sides of another triangle, then the triangles are similar.
• SAS (Side-Angle-Side) Similarity Theorem


  If two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar.
• The SSS and SAS Similarity Theorems require only that sides be proportional, not equal.
• Pythagorean Theorem


  The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs
• Proving Pythagorean Theorem


  1. Draw altitude CP to the hypotenuse

  2. Use leg rule in the similarity on a right triangle theorem

  3. Apply fundamental law of proportion

  4. Use addition property of equality

  5. Apply distributive property of multiplication over addition

  6. Use segment addition postulate

  7. Substitute to get c^2 = a^2 + b^2
• Right triangle with sides a=3, b=4, c=5

  • c^2 = a^2 + b^2
  • 25 = 9 + 16
  • 25 = 25, so it exists
• Right triangle with sides a=5, b=6, c=7

  • c^2 = a^2 + b^2
  • 49 ≠ 25 + 36, so it does not exist
• Right triangle with sides a=5, b=12, c=13

  • c^2 = a^2 + b^2
  • 169 = 144 + 25
  • 169 = 169, so it exists
• Shortest route problem

  • Given: a=3 dam, b=4 dam

  c^2 = a^2 + b^2

  c = √(9 + 16) dam

  c = 5 dam

  The shortest distance is 5 dam
• Pythagorean Triples


  • 3, 4, 5
  • 6, 8, 10
  • 9, 12, 15
  • 12, 16, 20
  • 15, 20, 25
  • 30, 40, 50
• Any triple that satisfies the Pythagorean relationship c^2 = a^2 + b^2 is a Pythagorean triple
• Pythagorean triple


  Any triple that satisfies the Pythagorean relationship c2 = a2 + b2
• Finding the shortest distance from house to school

  1. c2 = a2 + b2

  2. c = √(a2 + b2)

  3. c = 5 dam
• Table A - Pythagorean triples


  • a = 3, b = 4, c = 5
  • a = 6, b = 8, c = 10
  • a = ?, b = ?, c = ?
  • a = ?, b = ?, c = ?
  • a = ?, b = ?, c = ?
  • a = ?, b = ?, c = ?
• Table B - Pythagorean triples

  • a = 5, b = 12, c = 13
  • a = 10, b = 24, c = 26
  • a = ?, b = ?, c = ?
  • a = ?, b = ?, c = ?
  • a = ?, b = ?, c = ?
  • a = ?, b = ?, c = ?
• Pythagorean Theorem


  The square of the hypotenuse side is equal to the sum of the squares of the two other sides
• To determine if a triangle is right, compare if the square of the longest side is equal to the sum of the squares of the two other sides
• Any triple that satisfies the Pythagorean relationship c2 = a2 + b2 is a Pythagorean triple
• Solving the problem using the Pythagorean Theorem

  1. Given: TV screen dimensions 14 inches by 10 inches

  2. Find: Size of TV screen diagonal
• The size of a TV screen is given by the length of its diagonal
• Jerry wants to buy a 24-inch monitor

  Monitor found is 15 inches wide and 12 inches high
• Solving the problem using the Pythagorean Theorem

  1. Understand the problem

  2. Plan how to solve it

  3. Solve the problem

  4. Check the answer