PSAT Geometry — figures described in words, and the ratios that name their sides

Geometry on the real test usually comes with a drawing. This material has none, and every figure in it is described completely in words instead: the vertices are named, the right angle is placed at a named vertex, the correspondence between two similar triangles is spelled out, and the circle's measurement is stated as a radius or a diameter rather than left to be read off a picture. That is a genuine limitation of a text-only bank and also a useful exercise in its own right, because a candidate who can build the figure from a sentence has already done the hardest part of most of these questions. For anything that depends on reading a drawn diagram, use the College Board's own practice alongside this.

Twelve questions. Two are angles: a triangle whose third angle has to be found, and a transversal crossing parallel lines, where the wrong options are the supplement and the complement — both real angles in that arrangement, neither of them the one asked for.

Three are triangles. The Pythagorean theorem worked forwards on a six-eight-ten triangle, where the standard wrong answer adds the legs instead of squaring them. The theorem worked backwards, choosing which of four sets of three lengths makes a right triangle, including 10, 24 and 25, which misses by fifty-one and is there because 7, 24, 25 is the triple it resembles. And a pair of similar triangles with the vertex correspondence stated, so the only work is picking the right ratio and not inverting it.

Two are area and volume, including one where both dimensions of a rectangle double and the area therefore quadruples — with eight offered as an option, because that is what happens to a volume, and knowing which power applies to which is the whole point.

Two are circles, both left in terms of pi: a quarter sector of a circle of radius four, and the circumference of a circle given by its diameter, where using the radius in its place is the commonest error in the topic.

Two are right-triangle trigonometry — one asking which ratio sine is, the other asking for a tangent as a number, with the sine and the cosine offered beside it. And one is the distance between two points on the coordinate plane, which is the Pythagorean theorem again in different clothes.

Every figure and value here was invented for the exercise, and nothing is reproduced from any published test.

• Build a figure from a written description, naming the vertices and the right angle before starting
• Use the Pythagorean theorem in both directions, including testing whether three lengths make a right triangle
• Write the ratio between similar triangles the right way up, using the stated correspondence of vertices
• Apply the correct power when lengths are scaled: squared for an area, cubed for a volume
• Keep a circle answer in terms of pi, and check whether the number given is a radius or a diameter
• Name sine, cosine and tangent by the sides they use, and know that tangent never touches the hypotenuse
• Find a distance on the coordinate plane as the Pythagorean theorem in another form

There are no diagrams here; every figure is described in words inside its own question. Among them: triangle ABC with its right angle at C and legs of $6$ and $8$; a transversal crossing two parallel lines at $70^\circ$; triangle DEF similar to GHI with D corresponding to G, E to H and F to I; a rectangle of $5$ by $3$ with both dimensions doubled; a sector of a circle of radius $4$ cut by a central angle of $90^\circ$; a cylinder of radius $2$ and height $5$; a circle of diameter $10$; the sets $9, 40, 41$ and $10, 24, 25$ tested against the Pythagorean theorem; right triangle JKL with the right angle at K; right triangle PQR with the right angle at Q, legs $9$ and $12$; and the points $(1, 2)$ and $(4, 6)$. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the PSAT/NMSQT, and it is not an exam centre.