Geometry and Trigonometry

Pentru elevii români: Pe SAT primești o foaie de formule (reference sheet), așa că accentul cade pe aplicare, nu pe memorare; unghiurile pot apărea și în radiani, iar ecuația cercului se scrie $(x - h)^2 + (y - k)^2 = r^2$. Pentru trigonometrie, elevii americani folosesc mnemotehnica „SOH-CAH-TOA” pentru $\sin$, $\cos$ și $\tan$, iar segmentul se notează $\overline{AB}$.

What the SAT tests here

The Geometry and Trigonometry domain includes:

• area and volume (including composite shapes and scaling);
• lines, angles, and triangles (parallel lines, similar and congruent triangles);
• right triangles and trigonometry (Pythagorean theorem, special right triangles, sine, cosine, tangent);
• circles (arc length, sector area, central and inscribed angles, radians, and circle equations in the $xy$-plane).

The reference sheet in the testing app lists common area and volume formulas, the Pythagorean theorem, the special right triangles, and the facts that a circle has $360^\circ$ ($2\pi$ radians) and a triangle's angles sum to $180^\circ$. Know them anyway: looking them up costs time.

Key rules and formulas

Angles: angles on a line sum to $180^\circ$; vertical angles are equal; when a transversal crosses parallel lines, alternate interior angles are equal and same-side interior angles sum to $180^\circ$.

Similar triangles have equal corresponding angles and proportional sides. If the scale factor is $k$, areas scale by $k^2$ and volumes of similar solids by $k^3$.

Right triangles:

• $a^2 + b^2 = c^2$; common triples: 3-4-5, 5-12-13, 8-15-17.
• $45^\circ$-$45^\circ$-$90^\circ$: sides $s$, $s$, $s\sqrt{2}$.
• $30^\circ$-$60^\circ$-$90^\circ$: sides $s$, $s\sqrt{3}$, $2s$.

Trigonometry (SOH-CAH-TOA):

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}}$

Complementary angles: $\sin(x^\circ) = \cos(90^\circ - x^\circ)$.

Circles: circumference $C = 2\pi r$, area $A = \pi r^2$.

• Arc length $= \frac{\theta}{360^\circ} \cdot 2\pi r$, or $s = r\theta$ when $\theta$ is in radians.
• Sector area $= \frac{\theta}{360^\circ} \cdot \pi r^2$.
• Conversion: $180^\circ = \pi$ radians.
• Equation: $(x - h)^2 + (y - k)^2 = r^2$ with center $(h, k)$ and radius $r$.

Volumes: cylinder $\pi r^2 h$; cone $\frac{1}{3}\pi r^2 h$; sphere $\frac{4}{3}\pi r^3$; rectangular prism $\ell w h$; pyramid $\frac{1}{3}\ell w h$.

Worked examples

Example 1 (completing the square). Find the center and radius of the circle $x^2 + y^2 - 6x + 10y + 9 = 0$.

Group and complete the square: $(x^2 - 6x + 9) + (y^2 + 10y + 25) = -9 + 9 + 25$ $(x - 3)^2 + (y + 5)^2 = 25$ Center $(3, -5)$, radius $5$. Note the sign flip: $+5$ inside the parentheses means $k = -5$.

Example 2 (trigonometry). In right triangle $ABC$, angle $C$ is the right angle and $\sin A = \frac{5}{13}$. Find $\tan A$ and $\cos B$.

$\sin A = \frac{5}{13}$ means the side opposite $A$ is 5 and the hypotenuse is 13 (or a multiple). The other leg is $\sqrt{13^2 - 5^2} = 12$. So $\tan A = \frac{5}{12}$. Because $A$ and $B$ are complementary, $\cos B = \sin A = \frac{5}{13}$.

Example 3 (similar triangles). In triangle $ABC$, point $D$ lies on $\overline{AB}$ and point $E$ lies on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$. If $AD = 4$, $DB = 6$, and $DE = 5$, what is $BC$?

Parallel lines make triangle $ADE$ similar to triangle $ABC$. Corresponding sides: $\frac{AD}{AB} = \frac{DE}{BC}$, where $AB = 4 + 6 = 10$. So $\frac{4}{10} = \frac{5}{BC}$ and $BC = 12.5$. The trap is using $DB = 6$ instead of the whole side $AB = 10$.

Test-taking strategy

• Desmos defaults to radians. Before computing something like $\sin(35^\circ)$, open the graph settings (the wrench icon) and switch to degrees, or you will get the sine of 35 radians.
• Graph circles directly. Type the equation exactly as given, even in expanded form; Desmos draws the circle, and you can read the center and radius from the graph.
• Draw and label. If no figure is given, sketch one. If a figure is given, do not assume it is to scale unless the question says so.
• Look for special triangles. Side ratios such as $1 : \sqrt{3} : 2$ or a $45^\circ$ angle usually mean you can skip the calculator.
• Convert units of angle carefully. $\frac{2\pi}{3}$ radians $= 120^\circ$. Multiply radians by $\frac{180}{\pi}$ to get degrees.

Common traps

• Sign errors in circle equations: $(x + 2)^2$ means $h = -2$.
• Forgetting the square root: in $(x - h)^2 + (y - k)^2 = 49$, the radius is 7, not 49.
• Arc length versus sector area: arc length uses $2\pi r$; sector area uses $\pi r^2$.
• Using part of a side in similar-triangle proportions instead of the whole side.
• Linear scaling of area or volume: doubling every length multiplies area by 4 and volume by 8.
• Mixing degrees and radians in the formula $s = r\theta$, which needs radians.

Summary

Master angle relationships, similarity, the Pythagorean theorem with special right triangles, SOH-CAH-TOA with the complementary-angle identity, and circle facts in both degrees and radians. Complete the square to read a circle's center and radius. In Desmos, switch to degree mode for degree-based trigonometry, and graph circle equations to check your work.