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Standard IX

Mathematics

Euclidean Geometry

Let ABC be an...

Question

Let ABC be an acute-angled triangle. The circle $Γ$ with BC as diameter intersects AB and AC again at P and Q, respectively. Determine $∠$ BAC given that the orthocentre of triangle APQ lies on $Γ$ .

15^{o}

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25^{o}

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35^{o}

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45^{o}

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Solution

The correct option is D $45^{o}$
Let K denote the orthocenter of triangle APQ.
Since triangles ABC and AQP are similar it follows that K lies in the interior of triangle APQ.
Note that $∠ K P A = ∠ K Q A = 90^{o} - ∠ A$ .
Since BPKQ is a cyclic quadrilateral it follows that $∠$ BQK = 180 $^{o}$ - $∠$ BPK = 90 $^{o}$ - $∠$ A, while on the other hand $∠$ BQK = $∠$ BQA - $∠$ KQA = $∠$ A since BQ is perpendicular to AC.
$∴ 90^{\circ} - ∠ A = ∠ A$ ,
So $∠ A = 45^{o}$ .

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Let ABC be an acute-angled triangle. The circle Γ with BC as diameter intersects AB and AC again at P and Q, respectively. Determine ∠BAC given that the orthocentre of triangle APQ lies on Γ.

Let ABC be an acute-angled triangle. The circle $Γ$ with BC as diameter intersects AB and AC again at P and Q, respectively. Determine $∠$ BAC given that the orthocentre of triangle APQ lies on $Γ$ .