Solve by Element List Regression

Circle Geometry with Perpendicular Bisector on the SAT: Desmos Walkthrough

Use the equal-tangent rule and the right angle between a radius and a tangent, letting Desmos evaluate the distance formula to get the length.

This lesson uses sliders and direct evaluation in Desmos: put the unknown constant on a slider or a single expression line and let the calculator find the value instead of solving algebra by hand.

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A real SAT-style question

A circle has center $G$ , and points $M$ and $N$ lie on the circle. Line segments $M H$ and $N H$ are tangent to the circle at points $M$ and $N$ , respectively. If the radius of the circle is $168$ millimeters and the perimeter of quadrilateral $G M H N$ is $3,856$ millimeters, what is the distance, in millimeters, between points $G$ and $H$ ?

$168$

$1,752$

$1,760$

$1,768$

How to solve it in Desmos, step by step

The idea

Use the equal-tangent rule to get one tangent length, then the right angle between a radius and a tangent to find GH with the distance formula.

1
Step 1
Both radii GM and GN equal 168, and the two tangent segments from outside point H are equal, so the perimeter equals 168 plus 168 plus two times the tangent length. Type this line and read the value Desmos assigns to t, which is 1760, the length of one tangent segment.
Type in Desmos
$t=\frac{3856-2\cdot168}{2}$
2
Step 2
A tangent meets the radius at a right angle, so triangle GMH has a right angle at M with legs 168 and t. Type this distance line and read the result Desmos shows, which is 1768, matching choice D.
Type in Desmos
$d=\sqrt{168^{2}+t^{2}}$

The answer is D.

$1,768$

Now you try one

Same question type, different numbers. Use the exact Desmos moves from the walkthrough above.

A circle has center

P

, and points

R

and

S

lie on the circle. Line segments

RT

and

ST

are tangent to the circle at points

R

and

S

, respectively. If the radius of the circle is

144

millimeters and the perimeter of quadrilateral

PRTS

1{,}568

millimeters, what is the distance, in millimeters, between points

P

and

T

Practice this with a live Desmos calculator

The interactive version of this lesson builds the Desmos graph up step by step as you click through, with a live calculator you can type into and your progress saved.

Open the interactive lesson

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