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Standard X

Mathematics

Angle Subtended by an Arc of a Circle on the Circle and at the Center

In the figure...

Question

In the figure, diameter AB and chord QR are produced to meet at P. If $∠$ QPA $= 26^{o}$ , $∠$ QAR $= 36^{o}$ , find $^x$ and $^y$ .

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Solution

Given $∠ Q P A = 26, ∠ Q A R = 36$

$A B$ is a diameter

$∠ A Q B = 90$

$∠ B A R = ∠ B Q R = x$

In $△ A Q P$

$∠ Q A P + ∠ A P Q + ∠ A Q P = 180$

$(36 + x) + 26 + (90 + x) = 180$

$2 x = 28$

$x = 14$

$∠ A R P = 180 - ∠ A R Q = 180 - y$

In $△ A R P$

$∠ R A P + ∠ A P R + ∠ P R A = 180$

$x + 26 + (180 - y) = 180$

$y = x + 26 = 14 + 26 = 40^{\circ}$

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In the figure, diameter AB and chord QR are produced to meet at P. If ∠QPA=26o, ∠QAR=36o, find ^x and ^y.

Given ∠QPA=26,∠QAR=36 AB is a diameter ∠AQB=90 ∠BAR=∠BQR=x In △AQP ∠QAP+∠APQ+∠AQP=180 (36+x)+26+(90+x)=180 2x=28 x=14 ∠ARP=180−∠ARQ=180−y In △ARP ∠RAP+∠APR+∠PRA=180 x+26+(180–y)=180 y=x+26=14+26=40∘

In the figure, diameter AB and chord QR are produced to meet at P. If $∠$ QPA $= 26^{o}$ , $∠$ QAR $= 36^{o}$ , find $^x$ and $^y$ .