To find the slope of a tangent line to a circle, find the slope of the radius from the center to the point of tangency, then take its negative reciprocal. The radius and the tangent meet at a right angle, so their slopes multiply to -1. You can check any answer on this page when you graph circles on the College Board version of Desmos on our Desmos on the SAT page.
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Question banks: October 2026 Math question bank
Notes and practice need a free account: one click with Google, and you land straight on the lesson.
This page works through one tangent-slope question from TTA's question bank, a guided question you finish yourself, and the "a circle has center O, and points A and B lie on the circle" questions that test radii and arcs. Each Desmos check uses lines you can type exactly as written.
The tangent rule
A tangent line touches a circle at one point. The radius drawn from the center to that point meets the tangent at a right angle. For nonvertical lines, perpendicular slopes are negative reciprocals: if one slope is a/b, the other is -b/a.
Two special cases need care. A horizontal radius has slope 0, so its tangent is vertical and has undefined slope. A vertical radius has undefined slope, so its tangent is horizontal with slope 0.
Worked example: tangent slope (I do)
A circle in the xy-plane has its center at (-4, -8). Line k is tangent to this circle at the point (-8, -9). What is the slope of line k?
From our September 2026 SAT Math question bank. Choices: -4, -1/4, 1/4, 4.
Find the radius slope. The radius runs from the center (-4, -8) to the tangent point (-8, -9):
m = (-9 - (-8)) / (-8 - (-4)) = -1 / -4 = 1/4
Flip and change the sign. The tangent is perpendicular to that radius, so take the negative reciprocal of 1/4:
m(tangent) = -4
The answer key agrees: the correct choice is -4. The other choices are -1/4, 1/4 and 4.
Check it on Desmos
First find the radius squared from the center to the tangent point: (-8 - (-4))2 + (-9 - (-8))2 = 16 + 1 = 17. Then type these lines:
(x+4)^2+(y+8)^2=17graphs the circle.y=-4(x+8)-9graphs the line through (-8, -9) with slope -4.y=(1/4)(x+4)-8graphs the radius line through the center and the tangent point.
Line 2 should touch the circle at (-8, -9) and nowhere else, and it should cross line 3 at a right angle. By hand, substituting line 2 into the circle leaves one solution, x = -8, which is what "tangent" means. Use the graph as a check, not as your only evidence. A zoom level can make two slopes look perpendicular when they are not.
Why the common wrong answers appear
The choice 1/4 is the radius slope. It comes from correct arithmetic followed by one missing step. The choice -1/4 changes the sign but does not take the reciprocal. The choice 4 takes the reciprocal but misses the sign change. Write the words "negative reciprocal" before you calculate, and you avoid all three.
Guided example: find the missing coordinate (we do)
A circle in the xy-plane has its center at (3, 7). Line t is tangent to this circle at the point (a, -4), where a is a constant. The slope of line t is 5/4. What is the value of a?
From our November 2024 SAT Math question bank. Choices: -43/4, -29/5, 59/5, 67/4.
Step 1. The tangent slope is 5/4, so the radius slope is its negative reciprocal, -4/5.
Step 2. The radius runs from (3, 7) to (a, -4), so its slope is (-4 - 7) / (a - 3) = -11 / (a - 3).
Step 3 (your turn). Set -11 / (a - 3) equal to -4/5 and solve for a.
Check your answer: cross-multiplying gives 4(a - 3) = 55, so a - 3 = 55/4 and a = 67/4. The answer key says 67/4.
A circle has center O, and points A and B lie on the circle
This stem shows up with different letters, such as center Q or center P. It hands you one fact: OA and OB are both radii, so they have the same length. Two question types follow from it.
Angles. Triangle OAB is isosceles, so its two base angles are equal. In a question from our June 2025 SAT Math question bank, center O has points R and S on the circle and angle ROS is 116 degrees. The base angles share the remaining 180 - 116 = 64 degrees, so angle RSO is 32 degrees. The answer key says 32.
Arcs. An arc is a fraction of the whole circle, and that fraction is its central angle over 360 degrees. In a question from our November 2025 SAT Math question bank, arc AB measures 45 degrees and has length 4π units. Since 45/360 = 1/8, the circumference is 8 × 4π = 32π. Circumference is 2πr, so the radius is 16, which matches the answer key.
Your turn. A circle has center O, and points A and B lie on the circle. The measure of arc AB is 20 degrees, and the length of this arc is 4 inches. What is the circumference, in inches, of the circle? (From our November 2024 SAT Math question bank; choices 4, 8, 16 and 72.)
Answer: 72. Twenty degrees is 1/18 of 360 degrees, so the circumference is 18 × 4 = 72 inches.
Reading a circle equation
If the question gives a circle in standard form, (x-h)^2 + (y-k)^2 = r^2, the center is (h,k). Watch the signs: (x+4)^2 means the center's x-coordinate is -4. Once you have the center and tangent point, the same slope method applies.
If the equation is expanded, complete the square for x and for y. A question in our October 2024 SAT Math question bank gives x^2 - 4x + y^2 - 8y - 80 = 0 and says the circle is inscribed in a square. Completing the square gives (x-2)^2 + (y-4)^2 = 100, so the center is (2, 4) and the radius is 10. The square's side equals the diameter, 20, so the perimeter is 80, which matches the answer key. You can type the expanded equation into Desmos exactly as given to see the circle, then use the algebra for the exact numbers.
When algebra is faster
College Board's calculator policy says some Math questions are better done without a calculator, even though Desmos is available for the whole Math section. On a tangent-slope question, the algebra is two subtractions and a flip. Use Desmos after you have an answer, when you want to see the line touch the circle once.
Mistakes to avoid
- Stopping at the radius slope. On the worked example, 1/4 is a listed choice. Write "negative reciprocal" next to your radius slope before you look at the choices.
- Reading the center with the wrong signs.
(x+4)^2puts the center at x = -4, not 4. If your Desmos circle sits in the wrong quadrant, check the signs first. - Mixing up the arc fraction. The fraction is the central angle over 360. In the 20-degree arc question, dividing 360 by 20 gives 18, and 4 × 18 = 72. Multiplying 4 by 20 instead gives 80, which is not a choice.
Practice next
Use the free Circles lesson notes. On each tangent problem, write the radius slope before looking at the choices. For mixed SAT-style math sets, open the TTA SAT question bank.
A tangent line meets its circle at exactly one point, which makes it a system with one solution. See how to solve systems of equations on Desmos for that side of the idea. For more circle and geometry work, read the unit circle on Desmos and SAT volume problems on Desmos.
If the wording is the part that slows you down, review how to translate SAT math word problems. Create a free TTA account if you want to save your work and continue from the circle practice set.
Math checked by hand and in Python against our answer keys for all six bank questions on September 26, 2026. Keys re-checked on September 29, 2026. Calculator facts checked against College Board's calculator policy and Desmos's College Board calculator sheet. The Test Advantage is independent of College Board and Desmos.
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Frequently Asked Questions
How do you find the slope of a tangent line to a circle?
Find the slope of the radius from the center to the point of tangency, then take its negative reciprocal. With center (-4, -8) and tangency point (-8, -9), the radius slope is 1/4, so the tangent slope is -4.
Why is the tangent perpendicular to the radius?
A tangent touches the circle at one point, and at that point it meets the radius at a right angle. That is why the two slopes are negative reciprocals when neither line is vertical.
How do I check a tangent line on Desmos?
Type the circle equation, then type the line through the tangency point with your slope, for example (x + 4)^2 + (y + 8)^2 = 17 and y = -4(x + 8) - 9. The line should touch the circle at one point.
How do I read the center from a circle equation?
In (x + 4)^2 + (y + 8)^2 = r^2 the center is (-4, -8), because the signs flip. If the equation is expanded, complete the square for x and for y first.
A circle has center O, and points A and B lie on the circle. What does that tell you?
OA and OB are both radii, so triangle OAB is isosceles and its base angles are equal. If angle AOB is 116 degrees, each base angle is 32 degrees. An arc AB of 20 degrees is 1/18 of the circle, so a 4-inch arc means a 72-inch circumference.