In the figure, is the diameter of a circle. Point lies on the circle, and segment is perpendicular to at point . If and , which choice is the value of ? Note: Figure not drawn to scale.

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Super-Hard SAT Math Question 121 | Free SAT Prep | Aniko

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Question 121·200 Super-Hard SAT Math Questions·Geometry and Trigonometry

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In the figure, $PQ$ is the diameter of a circle. Point $R$ lies on the circle, and segment $RS$ is perpendicular to $PQ$ at point $S$ .

If $PQ=130$ and $PR=13\sqrt{5}$ , which choice is the value of $\frac{PQ}{PS}$ ?

Note: Figure not drawn to scale. an‍іk⁠o.аi

When a problem involves a diameter and a point on the circle, immediately look for a right triangle (the angle opposite the diameter is $90^\circ$ ). If an altitude is drawn to the hypotenuse, use the right-triangle similarity relationships; the fastest here is $\text{(leg)}^2=\text{(hypotenuse)}\times\text{(adjacent projection)}$ , which directly links $PR$ , $PQ$ , and $PS$ without extra computation. ©⁠ аnіk‌о.‍аi

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Use the diameter fact

What is the measure of an angle that subtends a diameter of a circle?

Connect $RS$ to a right-triangle property

Since $RS\perp PQ$ , point $S$ is where the altitude from the right angle meets the hypotenuse. аni‌кo.аi

Relate a leg to its projection

In a right triangle with altitude to the hypotenuse, a leg squared equals (hypotenuse) times (the adjacent segment of the hypotenuse). Try writing an equation using $PR$ , $PQ$ , and $PS$ .

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Store the given lengths

Enter PQ=130 and PR=13*sqrt(5).

Compute $PS$ using the right-triangle relationship

Enter PS=PR^2/PQ. Writt‍еn bу A‍ni‌ко

Compute the requested ratio

Enter PQ/PS and evaluate the value shown. Match that value to the answer choices.

Step-by-step Explanation

Use the diameter to identify a right triangle

Since $PQ$ is the diameter and $R$ is on the circle, $\angle PRQ$ is a right angle. Therefore, $\triangle PRQ$ is a right triangle with hypotenuse $PQ$ .

Apply the leg–projection relationship

Because $RS$ is perpendicular to $PQ$ , point $S$ is the foot of the altitude from the right angle to the hypotenuse.

In a right triangle, each leg is the geometric mean of the hypotenuse and the adjacent segment of the hypotenuse. For leg $PR$ : Аniкo Quеs⁠tion Bаnk

PR^2 = (PS)(PQ)

So,

PS = \frac{PR^2}{PQ}

Compute the ratio

Compute $PR^2$ :

PR^2 = (13\sqrt{5})^2 = 169\cdot 5 = 845

Then

PS = \frac{845}{130}

\frac{PQ}{PS} = \frac{130}{\frac{845}{130}} = \frac{130^2}{845} = \frac{16900}{845} = 20

Therefore, the correct choice is 20.

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