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In a triangle of base a , the ratio of t...

In a triangle of base `a ,` the ratio of the other two sides is `r(<1)dot` Show that the altitude of the triangle is less than or equal to `(a r)/(1-r^2)`

Text Solution

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Let ABC be a triangle with base BC=a and altitude AD=p, then

Area of `DeltaABC=(1)/(2)bc sin A`
Also, area of `DeltaABC=(1)/(2)ap`
`:. (1)/(2)ap =(1)/(2)bc sin A`
`rArr p=(ab sin A)/(a)`
`rArr =(abc sin A)/(a^(2))`
`rArr p=(abc sin A*(sin^(2)B-sin^(2)C))/(a^(2)(sin^(2)B-sin^(2)C))`
`=(abc sinA*sin (B+C)sin(B-c))/((b^(2)sin^(2)A-c^(2)sin^(2)A))`
` [ "Using sine rule", (a)/(sinA)=(b)/(sinB)=(c)/(sinC)]`
`=(abcsin^(2)A*sin(B-C))/((b-^(2)c^(2))*sin^(2)r^(2))=(ar sin(B-C))/(1-r^(2))`
`=(ab^(2) r sin (B-C))/(b^(2)-b^(2)r^(2))=( ar sin (B-C))/(1-r^(2))`
` rArr ple(ar)/(1-r^(3)) " " =[ :. sin (B-C)le 1]`
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