Properties of Right Triangles - SAT Math

180 degrees. This is a fundamental property of all triangles.

Line. Has no endpoints and extends infinitely in both directions.

Area = $\frac{1}{2} \times \text{base} \times \text{height}$. Multiply base by height, then divide by 2 for triangle area.

60 degrees. Triangle angles sum to $180°$: $40° + 80° + x = 180°$.

Adjacent angles. Share a common vertex and a common side but no interior points.

A triangle with two equal sides. Two equal sides create two equal base angles in the triangle.

The right angle (90 degrees). The largest angle is always opposite the longest side (hypotenuse).

They are equal. Angles formed by intersecting lines across from each other are congruent.

60 degrees. Since all angles are equal and sum to $180°$, each is $180° ÷ 3 = 60°$.

Obtuse triangle. Contains one angle greater than $90°$ but less than $180°$.

Triangle Sum Theorem. States that interior angles of any triangle sum to $180°$.

$\frac{1}{2} \times \text{base} \times \text{height}$. Uses the two perpendicular sides as base and height.

$\text{tan } A = \frac{4}{3}$. Tangent equals sine divided by cosine.

$b = 12$. Using $9^2 + b^2 = 15^2$, so $b^2 = 144$.

$c = 5$. Using $3^2 + 4^2 = 9 + 16 = 25$, so $c = 5$.

Sine. In a 30-60-90 triangle, $\sin(30°) = \frac{1}{2}$.

$7\text{√}2$. Hypotenuse equals leg times $\sqrt{2}$ in 45-45-90 triangle.

$\frac{1}{2}$. Standard trigonometric value for 60-degree angle.

Cosecant. Reciprocal trigonometric function of sine.

Diagonal. Connects two vertices that are not adjacent (next to each other).

Alternate interior angles are equal. When parallel lines are cut by a transversal, alternate interior angles are congruent.

24 square units. Area = $\frac{1}{2} \times 6 \times 8 = 24$ square units.

180 degrees. This is a fundamental property of triangles in Euclidean geometry.

$5\sqrt{3}$. In a 30-60-90 triangle, longer leg = hypotenuse × $\frac{\sqrt{3}}{2}$

90 degrees. Triangle angles sum to $180°$: $180° - 30° - 60° = 90°$.

Median. Connects a vertex to the midpoint of the opposite side.

They are supplementary. Consecutive interior angles on the same side of a transversal sum to $180°$.

360 degrees. The exterior angles of any polygon always sum to $360°$.

$a^2 + b^2 = c^2$. Relates the sides of a right triangle: legs squared equal hypotenuse squared.

Right triangle. Contains exactly one $90°$ angle.

Exterior angle = sum of two remote interior angles. An exterior angle equals the sum of the two non-adjacent interior angles.

Isosceles right triangle. Has two $45°$ angles and one $90°$ angle with two equal legs.

They are equal. When parallel lines are cut by a transversal, corresponding angles are congruent.

Two sides of equal length. Has exactly two sides of equal length and two equal angles.

Sum of all sides. Add the lengths of all three sides together.

Acute triangle. All three angles measure less than $90°$.

360 degrees. Any quadrilateral's interior angles sum to $360°$.

$\cos(2\theta) = \cos^2(\theta) - \sin^2(\theta)$. Formula for cosine of double angle.

$\cos(\frac{\theta}{2}) = \sqrt{\frac{1 + \cos(\theta)}{2}}$. Formula for cosine of half angle.

$\frac{1}{2}ab \sin C$. Uses two sides and the included angle between them.

1. At 0°, the point on the unit circle is (1,0), so cosine equals 1.

$\text{sin}(A + B) = \text{sin}A \text{cos}B + \text{cos}A \text{sin}B$. Formula for sine of sum of two angles.

$\sin(2\theta) = \sqrt{3}$. Using double angle formula with given values.

Cosecant (csc). Cosecant is defined as $\frac{1}{\sin}$.

$\text{tan } 60^\text{o} = \frac{\text{sin } 60^\text{o}}{\text{cos } 60^\text{o}} = \frac{\frac{\text{sqrt}(3)}{2}}{\frac{1}{2}} = \text{sqrt}(3)$. Using the definition $\tan = \frac{\sin}{\cos}$.

$\text{sin}(30^\text{o}) = \frac{1}{2}$. Using $\cos(90° - 30°) = \sin 30°$.

$\text{sin}^2\theta + \text{cos}^2\theta = 1$. Fundamental trigonometric identity derived from the unit circle.

1. In a 45-45-90 triangle, opposite equals adjacent.

$\frac{1}{2}$. In a 30-60-90 triangle, the shortest side is half the hypotenuse.

[-1, 1]. Sine values are bounded between -1 and 1.

$\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

$Volume = length \times width \times height$. Multiply all three dimensions together.

Volume = side³. Side length cubed for equal dimensions.

Volume = 288π. Using $\frac{4}{3}\pi \times 6^3 = 288\pi$.

Volume = $48\pi$. Using $\frac{1}{3}\pi \times 4^2 \times 9 = 48\pi$.

Area = $\frac{1}{2}$ × base × height. Half the product of base and height.

Area = 16. Square the side length: $4^2 = 16$.

Lateral Surface Area = 2πrh. Only the curved side surface, excluding bases.

$Area = \frac{1}{2} \times base \times height$. Half the product of base and perpendicular height.

Area = 25π. Use $\pi r^2$: $\pi \times 5^2 = 25\pi$.

Area = $\pi r^2$. $\pi$ times the square of the radius.

Area = 40. Using $8 \times 5 = 40$.

Volume = 120. Using $15 \times 8 = 120$.

Area = $\frac{\theta}{360}$ × πr². Fraction of full circle based on central angle.

Area = $\frac{1}{2} \times (base_1 + base_2) \times height$. Average of parallel bases times height.

Surface Area = 150. Using $6 \times 5^2 = 150$.

Volume = πr^2h. Base area times height: $\pi r^2 \times h$.

Volume = 60. Using $3 \times 4 \times 5 = 60$.

Volume = 64. Using $4^3 = 64$.

$Area = length \times width$. Multiply the two perpendicular sides of the rectangle.

Surface Area = 52. Using $2(2 \times 3 + 2 \times 4 + 3 \times 4) = 52$.

Lateral Surface Area = 80π. Using $2\pi \times 4 \times 10 = 80\pi$.

Surface Area = 2lw + 2lh + 2wh. Sum of areas of all six rectangular faces.

Volume = length × width × height. Product of all three dimensions.

Surface Area = 6 × $side^2$. Six faces, each with area of $side^2$.

Circumference = 14π. Using $2\pi \times 7 = 14\pi$.

Area = 28. Using $\frac{1}{2} \times (6 + 8) \times 4 = 28$.

Volume = 27. Cube the side length: $3^3 = 27$.

$Area = base \times height$. Base times the perpendicular height to that base.

Volume = 63π. Using $\pi \times 3^2 \times 7 = 63\pi$.

Area = base × height. Same as rectangle when base and height are perpendicular.

Area = πr². Pi times radius squared.

Volume = πr²h. Base area times height for circular cross-section.

Surface Area = 24π. Using $\pi \times 3(3 + 5) = 24\pi$.

Circumference = $2\pi r$. Two times pi times radius.

Surface Area = $πr(r + l)$. Base area plus lateral surface area.

$Volume = \frac{4}{3} \times πr^3$. Four-thirds times $\pi$ times radius cubed.

Area = 6π. Using $\frac{60}{360} \times \pi \times 6^2 = 6\pi$.

Area = length × width. The fundamental formula for rectangular area.

Area = 30. Using $\frac{1}{2} \times 10 \times 6 = 30$.

Surface Area = 2πr(h + r). Two circular bases plus lateral surface area.

Volume = $\frac{1}{3}$ × base area × height. One-third of prism volume with same base and height.

Area = 25π. Using $\pi \times 5^2 = 25\pi$.

Surface Area = 28π. Using $2\pi \times 2(5 + 2) = 28\pi$.

Volume = 20. Using $\frac{1}{3} \times 10 \times 6 = 20$.

Surface Area = 4πr². Four times the area of a great circle.

Volume = $\frac{4}{3} \pi r^3$. Four-thirds pi times radius cubed.

Volume = base area × height. Base area times perpendicular height.

Volume = $\frac{1}{3} \pi r^2 h$. One-third of cylinder volume with same base and height.

Surface Area = 36π. Using $4\pi \times 3^2 = 36\pi$.

Area = 21. Using $7 \times 3 = 21$.

$\frac{\text{sqrt}(3)}{2}$. Standard value for 30° in the unit circle.

1. Since $\tan 45° = 1$, cotangent is the reciprocal.

$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity derived from the Pythagorean theorem.

$\text{cos}(2\theta) = -\frac{1}{2}$. Using double angle formula with given values.

$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$. Formula for tangent of sum of two angles.

1. At 0°, the x-coordinate on the unit circle is 1.

$\pi$ radians. Use the conversion factor $\frac{\pi}{180}$.

$\text{cos}(60^\text{o}) = \frac{1}{2}$. Using $\sin(90° - 60°) = \cos 60°$.

1. At 0°, the adjacent side equals the hypotenuse.

$\frac{1}{2}$. Standard value for 60° in the unit circle.

$\text{tan}(2\theta) = \frac{2\text{tan}(\theta)}{1-\text{tan}^2(\theta)}$. Formula for tangent of double angle.

$2\pi$. Sine completes one cycle every $2\pi$ radians.

$\frac{1}{\sqrt{3}}$. Using the 30-60-90 triangle ratios.

$\sin(2\theta) = 2 \sin(\theta) \cos(\theta)$. Formula for sine of double angle.

1. At 90°, the x-coordinate on the unit circle is 0.

$\text{tan} \theta = \frac{\text{opp}}{\text{adj}}$. Tangent equals opposite side divided by adjacent side.

$\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

$\frac{\sqrt{3}}{2}$. Standard value for 60° in the unit circle.

$\text{sin}(90^\text{o} - \theta) = \text{cos}(\theta)$. Complementary angle relationship for sine.

[-1, 1]. Cosine values are bounded between -1 and 1.

$2\pi$. Cosine completes one cycle every $2\pi$ radians.

$\frac{\text{√}2}{2}$. In a 45-45-90 triangle, opposite and adjacent sides are equal.

$\tan(\frac{\theta}{2}) = \sqrt{\frac{1 - \cos(\theta)}{1 + \cos(\theta)}}$. Formula for tangent of half angle.

1. Square of $\tan 45° = 1$ is 1.

Cotangent (cot). Cotangent is defined as $\frac{1}{\tan}$.

$\frac{1}{2}$. Square of $\frac{\sqrt{2}}{2}$ equals $\frac{1}{2}$.

$\frac{\text{sqrt}(2)}{2}$. Standard value for 45° in the unit circle.

$\text{cos}(90^\text{o} - \theta) = \text{sin}(\theta)$. Complementary angle relationship for cosine.

$\text{π}$. Tangent completes one cycle every $\pi$ radians.

$\text{sin}^2 \theta + \text{cos}^2 \theta = 1$. Fundamental identity derived from the unit circle.

$\text{cos}(A + B) = \text{cos}A \text{cos}B - \text{sin}A \text{sin}B$. Formula for cosine of sum of two angles.

Cosecant (csc). Since $\sin\theta = \frac{1}{\csc\theta}$, cosecant is sine's reciprocal.

1. At 90°, the opposite side equals the hypotenuse in a right triangle.

1. Secant is the reciprocal of cosine; $\cos 60° = \frac{1}{2}$.

$\frac{1}{2}$. Standard value for 30° in the unit circle.

1. At 90°, the y-coordinate on the unit circle is 1.

Secant. Secant function is defined as $\sec \theta = \frac{1}{\cos \theta}$.

$\text{sin}(\frac{\theta}{2}) = \text{sqrt}(\frac{1 - \text{cos}(\theta)}{2})$. Formula for sine of half angle.

$r = 5$. Radius equals half the diameter.

A line segment with both endpoints on the circle. Distinguishes chord from other circle segments.

$r = 5$. Radius equals $\sqrt{25} = 5$.

$A = \frac{\theta}{360} \times \pi r^2$. Sector area equals the fraction of the circle times the total area.

$(x-h)^2 + (y-k)^2 = r^2$. Where $(h,k)$ is the center and $r$ is the radius.

$A = 16\pi$. Apply $A = \pi r^2$ with $r = 4$.

Center is $(0, 0)$, $r = 7$. Origin center with $r = \sqrt{49} = 7$.

$A = 16\pi$. Use $A = \pi r^2$ with $r = 4$.

$r = 10$. Solve $20\pi = 2\pi r$ to get $r = 10$.

Radius = $4$. In standard form $(x-h)^2 + (y-k)^2 = r^2$, the radius is $\sqrt{16} = 4$.

$\text{Arc Length} = r\theta$. Formula with angle $\theta$ in radians.

$x^2 + y^2 = 81$. Standard form equation with $r^2 = 81$.

Radius. Fundamental distance measurement in circles.

$r = 10$. Radius equals half the diameter.

$r = 6$. Solve $36\pi = \pi r^2$ to get $r = 6$.

A minor arc is less than 180°; a major arc is more. Classification based on arc's angular measure.

They are perpendicular at the point of tangency. Radius and tangent form $90°$ angle at contact point.

$C = 2\pi r$. Standard formula relating circumference to radius.

$L = \frac{\theta}{360} \times 2\pi r$. Arc length equals the fraction of the circle times the circumference.

$A = \frac{1}{2}r^2\theta$. Formula for sector area with angle in radians.

Half the measure of the intercepted arc. Inscribed angle theorem relates to intercepted arc.

Circumference. Standard term for circle's perimeter.

Center = $(5, -3)$. The center coordinates are $(h,k)$ from $(x-h)^2 + (y-k)^2 = r^2$.

$x^2 + y^2 = r^2$. Special case of standard form with center at origin.

Segment. Region between chord and its corresponding arc.

A line that touches a circle at exactly one point. Key property distinguishing tangent from secant lines.

$r = 8$. Radius equals $\sqrt{64} = 8$.

A line that intersects a circle at two points. Distinguishes secant from tangent lines.

$d = 2r$. Diameter is twice the radius.

Circles with the same center but different radii. Circles sharing center point with different sizes.

$A = \frac{3\pi}{2}$. Use $A = \frac{1}{2}r^2\theta$ with $\theta = \frac{\pi}{6}$.

Perpendicular bisector. Property of line from center to chord midpoint.

$C = 24\pi$. Apply $C = 2\pi r$ with $r = 12$.

$C = 14\pi$. Apply $C = \pi d$ with $d = 14$.

The diameter. Diameter passes through center, maximizing length.

Diameter = $10$ cm. Using $C = 2\pi r$: $31.4 = 2\pi r$, so $r = 5$, diameter = $10$.

$A = \pi r^2$. Standard formula for area using radius squared.

$C = 2\pi r$. The distance around a circle equals $2$ times $\pi$ times the radius.

$C = 14\pi$. Apply $C = 2\pi r$ with $r = 7$.

$Ax^2 + Ay^2 + Dx + Ey + F = 0$. The expanded form where $A$, $D$, $E$, and $F$ are constants.

Diameter. Standard term for longest chord through center.

Diameter = $14$ cm. Diameter equals twice the radius: $2 \times 7 = 14$ cm.

$d = 18$. Solve $18\pi = 2\pi r$ to get $r = 9$, so $d = 18$.

Center is $(3, -2)$. Center coordinates are $(h,k) = (3,-2)$.

$\text{Arc Length} = \frac{5\pi}{3}$. Use $s = r\theta$ where $\theta = \frac{\pi}{3}$ radians.

$\text{Arc Length} = 2\pi$. Use $s = r\theta$ with $\theta = \frac{\pi}{2}$.

$A = \pi r^2$. Area equals $\pi$ times the radius squared.

$(x - h)^2 + (y - k)^2 = r^2$. Standard form with center $ (h,k) $ and radius $ r $.

Secant (sec). Secant is defined as $\frac{1}{\cos}$.

1. Secant of 0 degrees equals 1 divided by cos 0.

$\frac{\text{opposite}}{\text{hypotenuse}}$. Ratio of side opposite angle to longest side.

$45^\text{o}$. Angle whose tangent equals 1 is 45 degrees.

1. Apply Pythagorean theorem: $\sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$.

Cosecant. Cosecant is the reciprocal of sine function.

1. Cotangent of 45 degrees equals cos 45 divided by sin 45.

1. Tangent equals 1 when opposite equals adjacent.