Represent the distance of -P- from sides BC- AC and AB respectively- thenxa-H3-xb-Hb-xc-Hc is always equal to

The correct option is C 1 We have,Δ =Δ _BPC+Δ _APC+Δ _APB ⇒=Δ =1/2ax_a+1/2bx_b+1/2cx_c Also,Δ=1/2aH_a=1/2bH_b=1/2cH_c ⇒Δ=Δ (x_a/H_a+x_b/H_b+x_c/H_c ) ⇒=x_a/H_ ...

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Standard XII

Mathematics

Area of Triangle

represent the...

Question

$In the adjacent figure, 'P' is any arbitrary interior of the triangle ABC,$ H_a, H_b, H_c $are the length of altitudes drawn from vertices A, B and C respectively. If$ x_a, x_b and x_c $represent the distance of 'P' from sides BC, AC and AB respectively, then$

$\frac{x_{a}}{H_{a}} + \frac{x_{b}}{H_{b}} + \frac{x_{c}}{H_{c}} is always equal to$

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Solution

The correct option is C
1

$We have, Δ = Δ_{B P C} + Δ_{A P C} + Δ_{A P B} \Rightarrow = Δ = \frac{1}{2} a x_{a} + \frac{1}{2} b x_{b} + \frac{1}{2} c x_{c} A l s o, Δ = \frac{1}{2} a H_{a} = \frac{1}{2} b H_{b} = \frac{1}{2} c H_{c} \Rightarrow Δ = Δ (\frac{x_{a}}{H_{a}} + \frac{x_{b}}{H_{b}} + \frac{x_{c}}{H_{c}}) \Rightarrow = \frac{x_{a}}{H_{a}} + \frac{x_{b}}{H_{b}} + \frac{x_{c}}{H_{c}} = 1$

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In the adjacent figure, 'P' is any arbitrary interior of the triangle ABC, H_a, H_b, H_c are the length of altitudes drawn from vertices A, B and C respectively. If x_a, x_b and x_c represent the distance of 'P' from sides BC, AC and AB respectively, then xaHa+xbHb+xcHcis always equal to

The correct option is C 1 We have,Δ=ΔBPC+ΔAPC+ΔAPB⇒=Δ=12axa+12bxb+12cxcAlso,Δ=12aHa=12bHb=12cHc⇒Δ=Δ(xaHa+xbHb+xcHc)⇒=xaHa+xbHb+xcHc=1