TSI Math Study Guide
Geometry Concepts
Geometry questions on the TSI may use relationships among angles, triangles, similar figures, transformations, and points on the coordinate plane. You may also need the Pythagorean theorem or basic right-triangle trigonometry.
Angle relationships
Complementary
Two angles that add to 90°. The complement of 37° is 90° − 37° = 53°.
Supplementary
Two angles that add to 180°. The supplement of 125° is 180° − 125° = 55°.
Vertical angles
When two lines cross, opposite angles are equal. Adjacent angles are supplementary: 180° − 68° = 112°.
Parallel lines and a transversal
Corresponding angles are equal; alternate interior angles are equal; same-side interior angles are supplementary.
Angles in a triangle
The interior angles of every triangle add to 180°.
The Pythagorean theorem
c is the hypotenuse — opposite the right angle, always the longest side.
Make sure c is the hypotenuse. A common error is putting the longest side in the wrong position.
Similar figures
Two similar triangles. 6 corresponds to 15; 8 corresponds to x.
Scale factor
15 ÷ 6 = 2.5 — every length in the larger figure is 2.5 times the corresponding length in the smaller. Match corresponding sides carefully.
Transformations
Translation
Slides the figure; size and shape unchanged.(x, y) → (x + 4, y + 2)
4 right, 2 up: (1, 3) → (5, 5)
Reflection
Flips the figure across a line.across the x-axis: (x, y) → (x, −y)across the y-axis: (x, y) → (−x, y)
(4, −3) across the x-axis → (4, 3)
Rotation
Turns the figure around a point, usually the origin.180° about the origin: (x, y) → (−x, −y)
(3, −5) → (−3, 5)
Dilation
Changes size, preserves shape. Centered at the origin with scale factor k:(x, y) → (kx, ky)
Scale factor 2: (3, 4) → (6, 8). The only one of the four that changes size.
Right-triangle trigonometry
SOH-CAH-TOA
sin θ = oppositehypotenusecos θ = adjacenthypotenusetan θ = oppositeadjacent
“Opposite” and “adjacent” depend on which acute angle you’re using.
Find tan θ
Opposite 6, adjacent 8.
Find a missing side
A 30° angle and a hypotenuse of 12. Find the side opposite 30°.
Order of operations for trig
Identify the hypotenuse first, then label the other two sides relative to the given angle.
Distance on the coordinate plane
Midpoint
Don’t mix them up
Midpoint means average the coordinates. Distance involves differences, squares, and a square root.
TSI strategy: draw or label the figure
If a problem is described only in words, make a quick sketch.
Right triangle
Mark the right angle, the hypotenuse, known lengths, and the missing side or angle.
Similar figures
Mark the corresponding sides.
Coordinate problems
Plot or roughly locate the points.
A simple diagram makes the relationship easier to recognize and keeps values from landing in the wrong places in a formula.
Don’t choose a formula just because you recognize the shape. First identify what information you have and what you need to find.
Side lengths of a right triangle
a2 + b2 = c2
An acute angle and side lengths
SOH-CAH-TOA
Similar figures
Proportions
Coordinate points
Distance, midpoint, or transformation rules