Similarity of Triangles PDF 2024

# Similarity of triangles ## Geometry 2024 ### Class ## In the opposite figure: AE and BC are two intersecting chords at D in a circle where D is the midpoint of BC * **Prove that:** * **1** AADC~A BDE * **2** (BD)$^2$ = AD × DE > ![A circle with two intersecting chords](./assets/img...

# Similarity of triangles ## Geometry 2024 ### Class ## In the opposite figure: AE and BC are two intersecting chords at D in a circle where D is the midpoint of BC * **Prove that:** * **1** AADC~A BDE * **2** (BD)$^2$ = AD × DE >  ## In the opposite figure: DEO is a triangle, m (∠O) = m (∠DXY), DX = YO = 3 cm. and DY = 5 cm. **Find the length of: XE** >  ## Corollary 1 If a line is drawn parallel to one side of a triangle and intersects the other two sides or the lines containing them, then the resulting triangle is similar to the original triangle. **In each of the following figures :** >  * If DE // BC and intersects AB and AC at D and E respectively, then ∆ABC ~ A ADE ## In the opposite figure : CE ∩ BD = {A}, BC // DE , BC = 5 cm. and DE = 2.5 cm. **1** Prove that: ∆ABC ~ AADE **2** Find the value of : X >  ## Corollary 2 In any right-angled triangle, the altitude to the hypotenuse separates the triangle into two triangles which are similar to each other and to the original triangle. **In the opposite figure :** If ∆ ABC is a right-angled triangle at A and AD ⊥ BC, then A DBA~A DAC ~ ∆ АВС >  * and it is left to the student to prove this corollary by using the previous postulate and its remarks. ## In the opposite figure : ABC is a right-angled triangle at B and BD ⊥ AC If AD = 4.5 cm. and DC = 8cm., **find the values of : X and y** >  ## Second case ## Theorem 1 **S.S.S. similarity theorem** If the side lengths of two triangles are in proportion, then the two triangles are similar. **In the opposite figure :** **Prove that:** * **1** The two coloured triangles are similar. * **2** BD bisects ∠ ABE >  ## Third case ## Theorem 2 **S.A.S. similarity theorem** If an angle of one triangle is congruent to an angle of another triangle and lengths of the sides including those angles are in proportion, then the triangles are similar. ABC is a triangle in which: AB = 6 cm. and BC = 9 cm. Let D be the midpoint of AB and HEBC such that BH = 2 cm. **Prove that:** * **1** ADBH ~ СВА * **2** ADHC is a cyclic quadrilateral. >