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A Peakscor full-length test in progress: Reading and Writing, Test A, Module 1 of 4, question 7 of 27, with the passage, four answer choices, a highlighter, a 24:18 timer and Flag, Back and Next buttons.

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Every question is tagged by section, domain, skill and difficulty, so you can practice exactly what you need. Full tests run like test day: two Reading & Writing modules, a break, then two Math modules.

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MathProblem-Solving and Data AnalysisHard

A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?

A80%
B96%Correct
C100%The trap
D104%

Check with $100: $100then$120then$96

  1. Turn each change into a multiplier.

    Up 20% means × 1.20. Down 20% means × 0.80.

  2. Apply them in order.

    1.20 × 0.80 = 0.96

  3. Read the answer.

    The final price is 96% of the original. That’s B.

  4. Why not 100%?

    The 20% cut is taken from the higher price, so it removes more than the 20% increase added.

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Every Peakscor full test follows this order. In the standard tests, the second module of each section adapts to how you did on the first.

  1. Reading & Writing · Module 1

    27 questions32 min

  2. Reading & Writing · Module 2

    27 questions32 min

    Adapts to how you did on Module 1.

  3. Break

    10 min

  4. Math · Module 1

    22 questions35 min

  5. Math · Module 2

    22 questions35 min

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98 questions · 2 hr 14 min of testing, plus the break

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The Daily, plus 36 practice questions every day: 12 easy, 12 medium and 12 hard, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too. No card needed.

Pro is $20 a month or $200 a year: unlimited practice with no daily limit, plus every full-length test and Challenge test. Peak is $29 a month or $280 a year: everything in Pro, plus the full Learn course and Peak Plan. Cancel anytime from your account.

98 questions across four modules, in the digital SAT’s order: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.

Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests use the hardest material throughout.

Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.

A quick run with three hearts. Each right answer builds your streak: the questions get harder as it grows, and your points multiply (×2 from three in a row, ×3 from six). A miss costs a heart and drops you back to easy. Best is your longest streak.

Every day at midnight. You get a fresh 12 easy, 12 medium and 12 hard.

No. Peakscor is an independent SAT practice tool, not affiliated with the College Board. Scores shown are estimates.

Your next point starts today.

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Circles · Lesson 3

Tangent lines

A tangent line touches a circle at exactly one point, the point of tangency. The key fact: a tangent is perpendicular to the radius drawn to the point of tangency. Whenever a figure shows a tangent, find the radius to the point where it touches, and mark a right angle there.

That right angle builds a right triangle from the center O, the point of tangency T and a point P on the tangent line. The radius OT and the tangent segment PT are the legs, and OP, opposite the right angle, is the hypotenuse: OT2+PT2=OP2OT^2 + PT^2 = OP^2. In the figure below, 82+152=64+225=289=1728^2 + 15^2 = 64 + 225 = 289 = 17^2.

The right angle works for angles too. The two acute angles of triangle OTP add up to 90°, so if angle TOP is 70°, angle OPT is 20°.

Two tangents from one point. From a point P outside a circle you can draw two tangent segments, PA and PB. They have equal lengths. The quadrilateral OAPB has right angles at A and B, so angle APB and angle AOB add up to 360°−90°−90°=180°360° - 90° - 90° = 180°.

Distance to the circle is not distance to the center. If segment OP crosses the circle at B, then OP=OB+BP=r+BPOP = OB + BP = r + BP.

A tangent line and the radius to the point of tangency81517OTP
A tangent line and the radius to the point of tangency A circle with center O. A horizontal line touches the top of the circle at point T and is tangent to it there. Point P is on the tangent line, outside the circle. OT is labelled 8, TP is labelled 15 and OP is labelled 17. A right-angle mark is at T.

Worked example Easy

A tangent line at point T610OTP
A tangent line at point T A circle with center O. A vertical line touches the circle at point T and is tangent to it there. Point P is on the tangent line, above T and outside the circle. Segment OT, a radius, is labelled 6, and segment OP is labelled 10.

In the figure shown, the line through P is tangent to the circle with center O at point T. If OT=6OT = 6 and OP=10OP = 10, what is the length of PT?

  1. 44
  2. 88Answer
  3. 1616
  4. 2342\sqrt{34}

How to solve it

  1. A tangent is perpendicular to the radius at the point of tangency, so angle OTP is a right angle.
  2. In right triangle OTP the hypotenuse is OP, opposite the right angle: 62+PT2=1026^2 + PT^2 = 10^2.
  3. PT2=100−36=64PT^2 = 100 - 36 = 64, so PT=8PT = 8.

Why each choice is right or wrong

  • A. Incorrect. This subtracts the lengths, 10−6=410 - 6 = 4. In a right triangle you subtract the squares: 100−36=64100 - 36 = 64, so PT=8PT = 8.
  • B. Correct. Angle OTP is 90°, so PT2=102−62=64PT^2 = 10^2 - 6^2 = 64 and PT=8PT = 8.
  • C. Incorrect. This adds the lengths, 6+10=166 + 10 = 16. A leg of a right triangle comes from the Pythagorean theorem, and it must be shorter than the hypotenuse, 10.
  • D. Incorrect. This treats OP as a leg instead of the hypotenuse: PT2=62+102=136PT^2 = 6^2 + 10^2 = 136, so PT=136=234PT = \sqrt{136} = 2\sqrt{34}. OP is opposite the right angle at T, so it is the hypotenuse.

Worked example Medium

Two tangent segments from point P46°x°OABP
Two tangent segments from point P A circle with center O. Point P is outside the circle, to the right. Segments PA and PB touch the circle at A and B and are tangent to it there. Radii OA and OB are drawn. Angle APB, at P, is marked 46°, and angle AOB, at O, is marked x°.

In the figure shown, PA and PB are tangent to the circle with center O at points A and B. Angle APB measures 46°. What is the value of xx?

  1. 4646
  2. 6767
  3. 9292
  4. 134134Answer

How to solve it

  1. Each tangent is perpendicular to the radius at its point of tangency, so angles OAP and OBP are both 90°.
  2. The angles of quadrilateral OAPB add up to 360°: x+90+90+46=360x + 90 + 90 + 46 = 360.
  3. So x=360−226=134x = 360 - 226 = 134.

Why each choice is right or wrong

  • A. Incorrect. This assumes angle AOB equals angle APB, 46°. They are opposite corners of quadrilateral OAPB, which has right angles at the other two corners, so the two angles add up to 180°.
  • B. Incorrect. 67° is angle AOP, half of angle AOB: OP splits the figure into two matching right triangles, and in triangle OAP, 90−23=6790 - 23 = 67. Angle AOB is twice that, 134°.
  • C. Incorrect. This doubles 46° to get 92, as if angle APB were an inscribed angle. P is outside the circle, so the inscribed angle rule doesn't apply. Use the right angles at A and B instead.
  • D. Correct. The tangents make right angles with the radii at A and B, so x=360−90−90−46=134x = 360 - 90 - 90 - 46 = 134.

Trap Putting the right angle in the wrong place

The right angle is at the point of tangency, between the radius and the tangent line, not at the outside point and not at the center. So the hypotenuse is the segment from the center to the outside point.

Trap Distance to the circle used as distance to the center

When a question gives the distance from an outside point to the circle, measured along a line through the center, add the radius to get the distance to the center. Using the shorter distance as the hypotenuse is a common slip.

Desmos When Desmos isn’t faster

Tangent questions turn on spotting the right angle, which happens on the figure, so do that part by hand.

Desmos is handy for the square root at the end. Type sqrt(100-36) and it shows 8. To compare an answer like 2342\sqrt{34} with a decimal, type 2sqrt(34) and Desmos shows about 11.66.

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