Geometry regents

Question 1

What is the Distance Formula?

Answer 1

√[(x₂ − x₁)² + (y₂ − y₁)²] – Use to find how far apart two points are.

Question 2

What is the Midpoint Formula?

Answer 2

((x₁ + x₂)/2, (y₁ + y₂)/2) – Use to find the middle point between two coordinates.

Question 3

What is the Slope Formula?

Answer 3

(y₂ − y₁)/(x₂ − x₁) – Use to find how steep a line is.

Question 4

What is the Equation of a Circle?

Answer 4

(x − h)² + (y − k)² = r² – Center is (h, k), radius is r.

Question 5

How do you prove lines are parallel?

Answer 5

Show they have the same slope.

Question 6

How do you prove a quadrilateral is a rectangle?

Answer 6

Show all angles are 90° or use slope to show consecutive sides are perpendicular.

Question 7

What is the formula for the area of a triangle?

Answer 7

(1/2) × base × height

Question 8

What is the formula for the area of a parallelogram?

Answer 8

base × height

Question 9

What is the formula for the area of a trapezoid?

Answer 9

(1/2) × (base₁ + base₂) × height

Question 10

What is the formula for the area of a circle?

Answer 10

π × radius²

Question 11

What is the Circumference of a Circle?

Answer 11

2 × π × radius

Question 12

What is the formula for Arc Length?

Answer 12

(θ/360) × 2πr – Part of the circle’s perimeter.

Question 13

What is the formula for Sector Area?

Answer 13

(θ/360) × πr² – Area of a “pizza slice” of the circle.

Question 14

What is a Tangent Line to a Circle?

Answer 14

Always perpendicular to the radius at the point of contact.

Question 15

What is the Inscribed Angle Theorem?

Answer 15

Inscribed angle = half the intercepted arc.

Question 16

What is the formula for Volume of a Prism?

Answer 16

Base Area × height – Works for rectangular and triangular prisms.

Question 17

What is the formula for Volume of a Cylinder?

Answer 17

πr²h – Circular base area times height.

Question 18

What is the formula for Volume of a Pyramid?

Answer 18

(1/3) × Base Area × height

Question 19

What is the formula for Volume of a Cone?

Answer 19

(1/3) × πr²h

Question 20

What is the formula for Volume of a Sphere?

Answer 20

(4/3) × π × r³

Question 21

What is the Surface Area of a Sphere?

Answer 21

4πr²

Question 22

What is the Surface Area of a Cylinder?

Answer 22

2πrh + 2πr² – Sides + top + bottom.

Question 23

What is the Surface Area of a Cone?

Answer 23

πrl + πr² – r = radius, l = slant height.

Question 24

What is the Dilation Rule?

Answer 24

(x, y) → (kx, ky) – Multiplies size, doesn’t change shape.

Question 25

What is the Translation Rule?

Answer 25

(x, y) → (x + a, y + b) – Moves the shape left/right/up/down.

Question 26

What is the formula for Rotation 90° CCW (around origin)?

Answer 26

(x, y) → (−y, x)

Question 27

What is the formula for Rotation 180°?

Answer 27

(x, y) → (−x, −y)

Question 28

What is the Reflection over x-axis?

Answer 28

(x, y) → (x, −y)

Question 29

What is the Reflection over y-axis?

Answer 29

(x, y) → (−x, y)

Question 30

What is Rigid Motion?

Answer 30

Keeps shape and size (translations, reflections, rotations).

Question 31

What is Similarity Transformation?

Answer 31

Changes size but keeps shape (like a dilation).

Question 32

What is the Pythagorean Theorem?

Answer 32

a² + b² = c² – For right triangles.

Question 33

What is the 45°-45°-90° Triangle Rule?

Answer 33

Legs are equal, hypotenuse = leg × √2

Question 34

What is the 30°-60°-90° Triangle Rule?

Answer 34

Hypotenuse = 2 × short leg, long leg = short leg × √3

Question 35

What is sin(θ)?

Answer 35

Opposite / Hypotenuse

Question 36

What is cos(θ)?

Answer 36

Adjacent / Hypotenuse

Question 37

What is tan(θ)?

Answer 37

Opposite / Adjacent

Question 38

What is the Interior Angle Sum of a Polygon?

Answer 38

(n − 2) × 180°, where n = number of sides

Question 39

What is One Interior Angle of a Regular Polygon?

Answer 39

[(n − 2) × 180°] / n

Question 40

What is One Exterior Angle of a Regular Polygon?

Answer 40

360° / n

Question 41

What does the Triangle Angle Sum Theorem state?

Answer 41

Angles of any triangle always add to 180°.

Question 42

What is the Exterior Angle of a Triangle?

Answer 42

Equals the sum of the two opposite interior angles.

Question 43

What are Vertical Angles?

Answer 43

Are always equal.

Question 44

What are Alternate Interior Angles (parallel lines)?

Answer 44

Are equal.

Question 45

What are Corresponding Angles (parallel lines)?

Answer 45

Are equal.

Question 46

What does the Isosceles Triangle Theorem state?

Answer 46

Two equal sides mean two equal angles.

Question 47

What is the Converse of Isosceles Theorem?

Answer 47

Two equal angles mean two equal sides.

Question 48

What does the Triangle Inequality Theorem state?

Answer 48

Two sides must add to more than the third side.