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Omni-MATH / A circle ω₁ of radius 15 intersects a circle ω₂ of radius 13 at points P and Q. Point A is on line PQ such that P is between A and Q. R and S are the points of tangency from A to …

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problem

A circle ω1\omega_{1} of radius 15 intersects a circle ω2\omega_{2} of radius 13 at points PP and QQ. Point AA is on line PQP Q such that PP is between AA and QQ. RR and SS are the points of tangency from AA to ω1\omega_{1} and ω2\omega_{2}, respectively, such that the line ASA S does not intersect ω1\omega_{1} and the line ARA R does not intersect ω2\omega_{2}. If PQ=24P Q=24 and ∠RAS\angle R A S has a measure of 90∘90^{\circ}, compute the length of ARA R.
Plain-text mathematical notation (without MathML)
A circle ω₁ of radius 15 intersects a circle ω₂ of radius 13 at points P and Q. Point A is on line PQ such that P is between A and Q. R and S are the points of tangency from A to ω₁ and ω₂, respectively, such that the line AS does not intersect ω₁ and the line AR does not intersect ω₂. If PQ=24 and ∠RAS has a measure of 90^(∘), compute the length of AR.
Original LaTeX notation
A circle $\omega_{1}$ of radius 15 intersects a circle $\omega_{2}$ of radius 13 at points $P$ and $Q$. Point $A$ is on line $P Q$ such that $P$ is between $A$ and $Q$. $R$ and $S$ are the points of tangency from $A$ to $\omega_{1}$ and $\omega_{2}$, respectively, such that the line $A S$ does not intersect $\omega_{1}$ and the line $A R$ does not intersect $\omega_{2}$. If $P Q=24$ and $\angle R A S$ has a measure of $90^{\circ}$, compute the length of $A R$.

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Source and history

task_id
13c3f6e0-e205-502b-a1ca-cc7cc76b86f1
task_key
test--13c3f6e0-e205-502b-a1ca-cc7cc76b86f1
task_revision_id
1
config / split
/ test

Official source

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