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Standard VI

Mathematics

Circumcircle

Draw a triang...

Question

Draw a triangle $A B C$ in which $A B = A C$ and a circle through $B$ touches $A C$ at $D$ and intersect $A B$ at $P$ , if $D$ is the mid-point of $A C$ , then prove that $A B = 4 A P$ .

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Solution

${(A D)}^{2} = A P . A B$
$\Rightarrow {(\frac{A B}{2})}^{2} = A P . A B [∵ A D = \frac{A C}{2}; A C = A B]$
$\Rightarrow \frac{{A B}^{2}}{4} = A P . A B$
$\Rightarrow A B = 4 A P$ proved

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Draw a triangle ABC in which AB=AC and a circle through B touches AC at D and intersect AB at P, if D is the mid-point of AC, then prove that AB=4AP.

(AD)2=AP.AB⇒(AB2)2=AP.AB[∵AD=AC2;AC=AB]⇒AB24=AP.AB⇒AB=4AP proved

Draw a triangle $A B C$ in which $A B = A C$ and a circle through $B$ touches $A C$ at $D$ and intersect $A B$ at $P$ , if $D$ is the mid-point of $A C$ , then prove that $A B = 4 A P$ .

${(A D)}^{2} = A P . A B$
$\Rightarrow {(\frac{A B}{2})}^{2} = A P . A B [∵ A D = \frac{A C}{2}; A C = A B]$
$\Rightarrow \frac{{A B}^{2}}{4} = A P . A B$
$\Rightarrow A B = 4 A P$ proved