Geometry Class 7: Similarity of Triangles

Instructor: Ravi Prakash

Key Topics

• Proving similarity of triangles
• Applications of similarity

Methods to Prove Similarity of Triangles

1. SSS Similarity

   • If the ratios of all corresponding sides of two triangles are equal, the triangles are similar.

   • Example: Triangles with sides 6, 11, 13 are similar to triangles with sides 3, 5.5, 6.5 because each corresponding side ratio is 2.

   • Formula for corresponding sides:

     [ \frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR} ]

2. AA (Angle-Angle) Similarity

   • If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
   • Example: If DE is parallel to BC in triangle ABC, then triangle ADE is similar to triangle ABC by AA similarity.
   • Important properties:
     • Corresponding sides are in the same ratio as the heights, medians, inradii, and circumradii of the triangles.
     • The ratio of areas of two similar triangles is the square of the ratio of corresponding sides.

3. SAS (Side-Angle-Side) Similarity

   • If the ratios of two corresponding sides of two triangles are equal and the included angles are equal, then the triangles are similar.

   • Example: Triangles with sides 10, 8 and included angle 30° are similar to triangles with sides 5, 4 and included angle 30°.

   • Formula for corresponding sides with included angle:

     [ \frac{AB}{PQ} = \frac{BC}{QR} ]

Properties of Similar Triangles

1. Ratios of corresponding sides, heights, medians, inradii, and circumradii are equal.
2. Ratios of areas of two similar triangles are equal to the square of the ratios of corresponding sides.
3. In right triangles, the altitudes intersect at points on the sides, and these intersecting points create similar sub-triangles.

Important Results and Usage

• If triangle ABC is right-angled and BD is the altitude to hypotenuse AC, triangles ABD, BDC, and ABC are all similar.

  • Formulas:

    [ AD = \frac{AB^2}{AC}, \quad CD = \frac{BC^2}{AC}, \quad BD = \frac{AB \cdot BC}{AC} ]

Example Problems and Solutions

1. Finding lengths in right triangles using similarity

   • Given triangle ABC with sides 6, 8, 10, and altitude BD:

     [ AD = \frac{6^2}{10} = 3.6, \quad CD = \frac{8^2}{10} = 6.4, \quad BD = \frac{6 \cdot 8}{10} = 4.8 ]

2. Finding elements within nested geometrical shapes

   • Example: Square inscribed in a triangle with sides 20, 34, 42.
   • Approach: Use properties of similar triangles and Pythagorean triplets to derive the lengths.

Finding Height Using Heron's Formula

• Area of triangle using Heron's formula (with sides 20, 34, 42):

  [ s = \frac{20 + 34 + 42}{2} = 48 ]

  Area, [A = \sqrt{48(48 - 20)(48 - 34)(48 - 42)} = 336 ]

  • Ensuring the correct height by reconciling the area formula with Pythagorean triplets.

This captures the key teaching points, examples, and methodologies Ravri Prakash emphasized in his lecture on similarity of triangles.