SAT Math — right triangles: the theorem and the three ratios

Right triangles are where the test puts its trigonometry, and the whole of it rests on two things: knowing which side is which relative to an angle, and knowing which of three ratios pairs the sides you have.

Three questions use the Pythagorean theorem, and the third of them is the one that matters. When both short sides are known the squares are added; when the hypotenuse is one of the two lengths given, its square is the total and the other square comes OUT of it. A learner who reaches for the familiar shape without noticing which side is the hypotenuse adds where they should subtract, and that answer sits among the options. Every triangle in the set is checked to satisfy the theorem genuinely, and the explanations use size as a second check: a hypotenuse is always longer than either short side and always shorter than their sum, which rules out several wrong options before any arithmetic.

Two questions ask only which ratio a situation needs, with nothing to calculate. Naming the hypotenuse first settles most of it, because the hypotenuse always faces the right angle. If the hypotenuse is one of the sides you know, the answer is a sine or a cosine depending on whether the other known side faces your angle or sits beside it; if the hypotenuse is not among them, the tangent is the only ratio available. The second question is built so that noticing the hypotenuse is absent answers it immediately.

Three more compute a ratio from three given sides, and their wrong options are not arbitrary: each is a real quantity about the same triangle. The cosine is offered against the sine, the tangent against both, and the inverted fraction against the correct one — and a sine greater than one is impossible, since the hypotenuse is the longest side, which is a check worth carrying.

Two questions put the work into a situation, a ladder against a wall and a surveyor measuring a flagpole, where the ratio has to be chosen before any number can be used. Each supplies the value of the ratio it needs, so no table or calculator is required and the reasoning stays where it belongs. Two more use the special triangles, with the side relationships stated in the question rather than assumed, and one of them is confirmed independently by the theorem.

Every triangle is described in words with its right angle named, so a reader with no picture can reconstruct it exactly. Every person and place is invented, and nothing is drawn from any official publication.

• Identify the hypotenuse as the side facing the right angle, and name the other two relative to a chosen angle
• Use the Pythagorean theorem in both directions, adding for a hypotenuse and subtracting for a shorter side
• Check a length against the rule that a hypotenuse exceeds either leg and falls short of their sum
• Choose between the sine, cosine and tangent by asking whether the hypotenuse is among the sides you know
• Compute a ratio from three given sides without inverting it or picking up the wrong pair
• Set up a real situation — a ladder, a flagpole — by choosing the ratio before touching the numbers
• Use the two special right triangles when their side relationships are supplied

Written for this catalogue, with no source document. The content is the right-triangle material named in the College Board's own public description of the Geometry and Trigonometry domain — the Pythagorean theorem and right-triangle trigonometry; every question, option and explanation here is original, and nothing is reproduced from any official publication or released test. Note that the digital test presents most of this material with a figure on screen and allows a calculator; here every triangle is described in words and the value of any ratio needed is supplied in the question. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the SAT, and it is not an exam centre.