Geometric Mean of Similar Triangles 

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Geometric Mean of Similar Triangles : The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.


ΔABC ~ ΔBDC ~ ΔADB
Since these triangles are similar, we can establish proportions relating the corresponding sides.
Two valuable theorems can be formed using these proportions.

Given : A right triangle ABC, right angled at B. BD ⊥ AC.

Prove that : ΔABC ~ ΔBDC ~ ΔADB

Statements
Reasons
1) ∠ABC= 90 1) Given
2) BD ⊥ AC 2) Given
3) ∠BDC = ∠BDA = 900 3) By definition of perpendicular
4) ∠ABD + ∠DBC = 90 4) Since ∠ABC = 90
5) ∠C + ∠DBC + 90 = 180 5) Angle sum property
6) ∠C + ∠DBC = 90 6) By subtraction property
7) ∠C = ∠ABD 7) From (4) and (7)
8) ΔADB ~ Δ BDC 8) By AA postulate
( from 3 and 7)
9) ∠ADB = ∠ABC 9) Each 900
10) ∠A = ∠A 10) Reflexive (common angle)
11) ΔADB ~ ΔABC 11) By AA postulate
(from 9 and 12)
12) ∠C = ∠C 12) Reflexive (common angle )
13) ΔBDC ~ ΔABC 13) By AA postulate
( from 11 and 12)
14) ΔADB ~ Δ BDC 14) from (3) and (7)

Altitude Rule
The altitude to the hypotenuse is the mean proportional between the segments of hypotenuse.
x/h = h/y
⇒ h2 = xy

Leg Rule
Either leg of the given right triangle is the mean proportional between the hypotenuse of the given right triangle and the segment of the hypotenuse adjacent to that leg.

y/a = a/c ⇒ a2 = yc and x/b = b/c ⇒ b2= xc



Sample problems on Geometric mean of similar triangles


Problem 1 : If AD = 3 and DB = 9 find CD.


Solution :
(CD)2 = AD . DB

x2 = 3 x 9

x = ± √3 √9

x = ± 3√3

CD = 3√3 ( CD can not be negative , so reject -3√3)

Problem 2 : If AD = 3 and DB = 9 find AC.


Solution :
(AC)2 = AD . AB

y2 = 3 x 12 ( AB = AD + DB ; AB = 3 + 9 =12)

y2 = 36

y = ± 6

AC = 6 ( AC can not be negative , so reject -6 )
Similarity in Triangles

Similarity in Geometry
Properties of similar triangles
Basic Proportionality Theorem(Thales theorem)
Converse of Basic Proportionality Theorem
Interior Angle Bisector Theorem
Exterior Angle Bisector Theorem
Proofs on Basic Proportionality
Criteria of Similarity of Triangles
Geometric Mean of Similar Triangles
Areas of Two Similar Triangles

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