Title: Geometry Basics — LearnCoachAssist

Q: What is a point in geometry?
A: A point is a location in space that has no size, no width, no length, and no depth. It is represented by a dot and named with a capital letter.

Q: What is a line in geometry?
A: A line is a straight one-dimensional figure that extends infinitely in both directions. It has no thickness and is defined by any two points on it.

Q: What is a ray?
A: A ray is a part of a line that has one fixed endpoint and extends infinitely in one direction. It is named by its endpoint and another point on the ray.

Q: What is a line segment?
A: A line segment is a part of a line bounded by two distinct endpoints. Unlike a line, it has a definite length that can be measured.

Q: What is a plane in geometry?
A: A plane is a flat, two-dimensional surface that extends infinitely in all directions. It is defined by three non-collinear points.

Q: What does it mean for points to be collinear?
A: Points are collinear if they all lie on the same straight line. Any two points are always collinear, but three or more points may or may not be.

Q: What does it mean for points to be coplanar?
A: Points are coplanar if they all lie on the same plane. Any three points are always coplanar, but four or more points may or may not be.

Q: What is the difference between a convex and a concave shape?
A: A convex shape has all interior angles less than 180°, and any line segment between two interior points stays inside the shape. A concave shape has at least one interior angle greater than 180°, creating an indentation.

Q: What is an acute angle?
A: An acute angle is an angle that measures greater than 0° and less than 90°. Examples include 30°, 45°, and 60°.

Q: What is an obtuse angle?
A: An obtuse angle is an angle that measures greater than 90° and less than 180°. An example is 120°.

Q: What is a right angle?
A: A right angle measures exactly 90°. It is often indicated by a small square symbol at the vertex.

Q: What are complementary angles?
A: Complementary angles are two angles whose measures add up to exactly 90°. For example, 35° and 55° are complementary.

Q: What are supplementary angles?
A: Supplementary angles are two angles whose measures add up to exactly 180°. For example, 110° and 70° are supplementary.

Q: What are vertical angles?
A: Vertical angles are the pairs of opposite angles formed when two lines intersect. They are always congruent (equal in measure).

Q: What is a straight angle?
A: A straight angle measures exactly 180° and forms a straight line. It is essentially a half-rotation.

Q: What are adjacent angles?
A: Adjacent angles are two angles that share a common vertex and a common side but do not overlap. They lie next to each other.

Q: What are the three types of triangles classified by side length?
A: Equilateral (all three sides equal), isosceles (exactly two sides equal), and scalene (no sides equal). Each type also has corresponding angle properties.

Q: What are the three types of triangles classified by angle?
A: Acute (all angles less than 90°), right (one angle exactly 90°), and obtuse (one angle greater than 90°).

Q: What is the Triangle Angle Sum Theorem?
A: The sum of the three interior angles of any triangle is always 180°. This is a fundamental property used to find unknown angles in a triangle.

Q: What is the SSS congruence postulate?
A: The Side-Side-Side (SSS) postulate states that if all three sides of one triangle are congruent to the corresponding three sides of another triangle, then the two triangles are congruent.

Q: What is the SAS congruence postulate?
A: The Side-Angle-Side (SAS) postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

Q: What is the ASA congruence postulate?
A: The Angle-Side-Angle (ASA) postulate states that if two angles and the included side of one triangle are congruent to two angles and the included side of another, the triangles are congruent.

Q: What is the AAS congruence theorem?
A: The Angle-Angle-Side (AAS) theorem states that if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Q: What does it mean for two triangles to be similar?
A: Two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. Similar triangles have the same shape but may differ in size.

Q: What is the AA similarity postulate?
A: The Angle-Angle (AA) similarity postulate states that if two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Q: What is the Triangle Inequality Theorem?
A: The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This must hold for all three combinations.

Q: What is the Exterior Angle Theorem for triangles?
A: The measure of an exterior angle of a triangle equals the sum of the two non-adjacent interior angles. For example, if the remote interior angles are 40° and 60°, the exterior angle is 100°.

Q: What is the Pythagorean Theorem?
A: The Pythagorean Theorem states that in a right triangle, a² + b² = c², where a and b are the legs and c is the hypotenuse. It only applies to right triangles.

Q: In a right triangle with legs 3 and 4, what is the hypotenuse?
A: Using a² + b² = c²: 3² + 4² = 9 + 16 = 25, so c = √25 = 5. This is the most well-known Pythagorean triple: (3, 4, 5).

Q: What is a Pythagorean triple?
A: A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), and (8, 15, 17).

Q: What is the converse of the Pythagorean Theorem?
A: If the square of the longest side of a triangle equals the sum of the squares of the other two sides (c² = a² + b²), then the triangle is a right triangle.

Q: How can you determine if a triangle is acute or obtuse using side lengths?
A: If c² < a² + b², the triangle is acute. If c² > a² + b², the triangle is obtuse. Here c is the longest side.

Q: What are the side ratios in a 45-45-90 triangle?
A: In a 45-45-90 triangle, the sides are in the ratio 1 : 1 : √2. If each leg has length x, the hypotenuse is x√2.

Q: What is the formula for the circumference of a circle?
A: The circumference is C = 2πr or equivalently C = πd, where r is the radius and d is the diameter.

Q: What is the formula for the area of a circle?
A: The area of a circle is A = πr², where r is the radius. This can also be written as A = π(d/2)² using the diameter.

Q: What is a chord of a circle?
A: A chord is a line segment whose endpoints both lie on the circle. The longest chord in any circle is the diameter.

Q: What is a tangent line to a circle?
A: A tangent is a line that touches the circle at exactly one point, called the point of tangency. A tangent is always perpendicular to the radius drawn to the point of tangency.

Q: What is a secant of a circle?
A: A secant is a line that intersects a circle at exactly two points. Unlike a chord, a secant extends beyond the circle in both directions.

Q: What is the arc length formula?
A: The arc length is s = (θ/360°) × 2πr for angle θ in degrees, or s = rθ when θ is in radians. It represents the distance along the curved part of the circle.

Q: What is a central angle of a circle?
A: A central angle is an angle whose vertex is at the center of the circle and whose sides are radii. The measure of the central angle equals the measure of its intercepted arc.

Q: What is an inscribed angle, and how does it relate to a central angle?
A: An inscribed angle has its vertex on the circle and its sides are chords. It measures exactly half the central angle that subtends the same arc: inscribed angle = ½ × central angle.

Q: What is the formula for the area of a sector?
A: The area of a sector is A = (θ/360°) × πr² for angle θ in degrees. It represents the fraction of the circle's area enclosed by two radii and their intercepted arc.

Q: What is the relationship between an inscribed angle and its intercepted arc?
A: An inscribed angle is always half the measure of its intercepted arc. If the intercepted arc is 100°, the inscribed angle is 50°.

Q: What is a regular polygon?
A: A regular polygon is a polygon with all sides equal in length and all interior angles equal in measure. Examples include equilateral triangles, squares, and regular hexagons.

Q: What is the formula for the sum of interior angles of a polygon?
A: The sum of interior angles of a polygon with n sides is (n − 2) × 180°. For example, a hexagon (n=6) has interior angles summing to 720°.

Q: What is each interior angle of a regular polygon?
A: Each interior angle of a regular n-sided polygon is (n − 2) × 180° / n. For a regular pentagon, this gives (5−2)×180°/5 = 108°.

Q: What is the sum of exterior angles of any convex polygon?
A: The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides.

Q: How many diagonals does a polygon with n sides have?
A: The number of diagonals is n(n − 3) / 2. For example, a hexagon has 6(6−3)/2 = 9 diagonals.

Q: What is an apothem of a regular polygon?
A: The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It is used in the area formula: A = ½ × perimeter × apothem.

Q: What is the formula for the area of a regular polygon?
A: The area of a regular polygon is A = ½ × p × a, where p is the perimeter and a is the apothem.

Q: What is the formula for the area of a rectangle?
A: The area of a rectangle is A = length × width. The perimeter is P = 2(length + width).

Q: What is the formula for the area of a triangle?
A: The area of a triangle is A = ½ × base × height, where the height is the perpendicular distance from the base to the opposite vertex.

Q: What is the formula for the area of a parallelogram?
A: The area of a parallelogram is A = base × height, where the height is the perpendicular distance between the two parallel bases. The perimeter is P = 2(a + b).

Q: What is the formula for the area of a trapezoid?
A: The area of a trapezoid is A = ½ × (b₁ + b₂) × h, where b₁ and b₂ are the two parallel bases and h is the height.

Q: What is the formula for the area of a rhombus using its diagonals?
A: The area of a rhombus is A = ½ × d₁ × d₂, where d₁ and d₂ are the lengths of the two diagonals.

Q: What is Heron's formula for the area of a triangle?
A: Heron's formula is A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 is the semi-perimeter. It allows you to find the area using only the three side lengths.

Q: What is the perimeter of a regular polygon?
A: The perimeter of a regular polygon is P = n × s, where n is the number of sides and s is the length of each side.

Q: What is the volume of a rectangular prism (box)?
A: The volume is V = length × width × height. The surface area is SA = 2(lw + lh + wh).

Q: What is the volume of a cylinder?
A: The volume of a cylinder is V = πr²h, where r is the radius of the base and h is the height.

Q: What is the surface area of a cylinder?
A: The total surface area is SA = 2πr² + 2πrh. The first term is the area of the two circular bases, and the second is the lateral (side) surface area.

Q: What is the volume of a cone?
A: The volume of a cone is V = ⅓πr²h, where r is the radius of the base and h is the height. It is exactly one-third the volume of a cylinder with the same base and height.

Q: What is the surface area of a cone?
A: The total surface area is SA = πr² + πrl, where r is the base radius and l is the slant height. The slant height can be found using l = √(r² + h²).

Q: What is the volume of a sphere?
A: The volume of a sphere is V = (4/3)πr³, where r is the radius.

Q: What is the surface area of a sphere?
A: The surface area of a sphere is SA = 4πr². This is exactly four times the area of a great circle of the sphere.

Q: What is the volume of a pyramid?
A: The volume of a pyramid is V = ⅓ × base area × height. For a square pyramid with base side s, this becomes V = ⅓s²h.

Q: What is the volume of a triangular prism?
A: The volume is V = base area × length, where the base area is the area of the triangular cross-section. If the triangle has base b and height h, then V = ½bh × l.

Q: What is the volume of a hemisphere?
A: The volume of a hemisphere is V = (2/3)πr³, which is exactly half the volume of a full sphere.

Q: What is the surface area of a hemisphere (including the base)?
A: The total surface area is SA = 3πr². This includes the curved surface (2πr²) plus the flat circular base (πr²).

Q: What is the distance formula in coordinate geometry?
A: The distance between two points (x₁, y₁) and (x₂, y₂) is d = √[(x₂−x₁)² + (y₂−y₁)²]. It is derived from the Pythagorean Theorem.

Q: What is the midpoint formula?
A: The midpoint of a segment with endpoints (x₁, y₁) and (x₂, y₂) is M = ((x₁+x₂)/2, (y₁+y₂)/2). It gives the point exactly halfway between the two endpoints.

Q: What is the slope formula?
A: The slope of a line through points (x₁, y₁) and (x₂, y₂) is m = (y₂−y₁)/(x₂−x₁). Slope measures the steepness and direction of a line.

Q: What is the slope-intercept form of a line?
A: The slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept (the point where the line crosses the y-axis).

Q: What is the point-slope form of a line?
A: The point-slope form is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a known point on the line. It is useful when you know the slope and one point.

Q: What is the standard form of a linear equation?
A: The standard form is Ax + By = C, where A, B, and C are integers, and A is typically non-negative. It is useful for finding intercepts quickly.

Q: How do you find the equation of a line given two points?
A: First calculate the slope m = (y₂−y₁)/(x₂−x₁), then substitute m and one of the points into point-slope form: y − y₁ = m(x − x₁). Simplify to your preferred form.

Q: What is the formula for the area of a triangle given coordinates of its vertices?
A: The area is A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. This is known as the Shoelace formula for three vertices.

Q: What is the equation of a circle in standard form?
A: The standard form is (x−h)² + (y−k)² = r², where (h, k) is the center and r is the radius.

Q: How do you find the distance from a point to a line?
A: The distance from point (x₀, y₀) to line Ax + By + C = 0 is d = |Ax₀ + By₀ + C| / √(A² + B²). This gives the shortest (perpendicular) distance.

Q: What is a translation in geometry?
A: A translation slides every point of a figure the same distance in the same direction. It preserves size, shape, and orientation. In coordinates, it maps (x, y) to (x + a, y + b).

Q: What is a reflection in geometry?
A: A reflection flips a figure over a line called the line of reflection, creating a mirror image. It preserves size and shape but reverses orientation.

Q: What is a rotation in geometry?
A: A rotation turns a figure around a fixed point called the center of rotation by a specified angle. It preserves size, shape, and the distance of each point from the center.

Q: What is a dilation in geometry?
A: A dilation is a transformation that enlarges or reduces a figure by a scale factor k from a center point. It maps (x, y) to (kx, ky) when centered at the origin.

Q: What is the rule for reflecting a point over the x-axis?
A: Reflecting the point (x, y) over the x-axis gives (x, −y). The x-coordinate stays the same and the y-coordinate changes sign.

Q: What is the rule for reflecting a point over the y-axis?
A: Reflecting the point (x, y) over the y-axis gives (−x, y). The y-coordinate stays the same and the x-coordinate changes sign.

Q: What is the rule for rotating a point 90° counterclockwise about the origin?
A: Rotating (x, y) by 90° counterclockwise about the origin gives (−y, x). For 90° clockwise, the result is (y, −x).

Q: What are parallel lines?
A: Parallel lines are lines in the same plane that never intersect, no matter how far they are extended. They have the same slope.

Q: What are perpendicular lines?
A: Perpendicular lines are lines that intersect at a 90° angle. The product of their slopes is −1: if one slope is m, the other is −1/m.

Q: What are corresponding angles formed by a transversal?
A: Corresponding angles are on the same side of the transversal and in the same position relative to the parallel lines. When the lines are parallel, corresponding angles are congruent.

Q: What are alternate interior angles?
A: Alternate interior angles are on opposite sides of the transversal and between the parallel lines. When the lines are parallel, they are congruent.

Q: What are alternate exterior angles?
A: Alternate exterior angles are on opposite sides of the transversal and outside the parallel lines. When the lines are parallel, they are congruent.

Q: What are co-interior (same-side interior) angles?
A: Co-interior angles (also called consecutive interior or same-side interior angles) are on the same side of the transversal and between the parallel lines. They are supplementary, adding up to 180°.

Q: How can you prove two lines are parallel using angles?
A: Two lines cut by a transversal are parallel if any of these hold: corresponding angles are congruent, alternate interior angles are congruent, alternate exterior angles are congruent, or co-interior angles are supplementary.

Q: What is sine in a right triangle?
A: The sine of an angle θ is the ratio of the length of the side opposite the angle to the hypotenuse: sin θ = opposite / hypotenuse.

Q: What is cosine in a right triangle?
A: The cosine of an angle θ is the ratio of the length of the adjacent side to the hypotenuse: cos θ = adjacent / hypotenuse.

Q: What is tangent in a right triangle?
A: The tangent of an angle θ is the ratio of the opposite side to the adjacent side: tan θ = opposite / adjacent. It can also be expressed as sin θ / cos θ.

Q: What is the mnemonic SOH-CAH-TOA?
A: SOH-CAH-TOA helps remember the trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Q: What are the side ratios of a 30-60-90 triangle?
A: The sides are in the ratio 1 : √3 : 2. The side opposite 30° is the shortest, opposite 60° is √3 times the shortest, and the hypotenuse is twice the shortest.

Q: What is the Pythagorean identity in trigonometry?
A: The Pythagorean identity states sin²θ + cos²θ = 1 for any angle θ. It is derived from the Pythagorean Theorem applied to the unit circle.

Q: What are the reciprocal trigonometric functions?
A: The reciprocal functions are: cosecant csc θ = 1/sin θ, secant sec θ = 1/cos θ, and cotangent cot θ = 1/tan θ.

Q: How do you find an unknown side of a right triangle using trigonometry?
A: Identify the angle and which sides are involved (opposite, adjacent, hypotenuse), choose the appropriate ratio (sin, cos, or tan), set up the equation, and solve for the unknown side. For example, if you know angle θ and the hypotenuse, use opposite = hypotenuse × sin θ.