Special Numbers and Triangle Geometry Quiz

By Basic Proportionality Theorem (Thales theorem), if DE is parallel to BC, then AD/AB = AE/AC = DE/BC. Given AD:AB = 3:5, therefore DE:BC = 3:5.

The symbol ~ (tilde) is used in mathematics to denote similarity between geometric figures. For example, triangle ABC ~ triangle DEF means they are similar. Equality is denoted by =.

By the Basic Proportionality Theorem, if PQ || YZ, then XP/XY = XQ/XZ. Given XP/XY = 2/3, then XQ/XZ = 2/3. This means XQ is 2 parts and XZ is 3 parts, leaving QZ as 1 part. Thus, XQ:QZ = 2:1.

This is the Basic Proportionality Theorem (also called Thales' theorem). When a line divides two sides of a triangle in the same ratio, it must be parallel to the third side. This creates similar triangles.

The converse of the Midpoint Theorem states that a line drawn through the midpoint of one side of a triangle, parallel to another side, bisects the third side. The statement provided in the question is not the standard converse.

For similar triangles, the ratio of areas is the square of the ratio of ANY corresponding linear dimensions - sides, altitudes, medians, or perimeters. However, the most fundamental and commonly stated version uses the sides directly.

All equilateral triangles have internal angles of 60 degrees. Since their corresponding angles are always equal, they satisfy the AA similarity criterion, making all equilateral triangles similar regardless of side length.

In triangles BDC and CEB, both have a 90-degree angle (at D and E), share the base BC, and have equal base angles (angle C = angle B) because triangle ABC is isosceles. By AA similarity, BDC ~ CEB.

This is the SSS (Side-Side-Side) similarity criterion. If all corresponding sides are in proportion, the triangles must be similar - their corresponding angles are automatically equal.

The product of two numbers is equal to the product of their HCF and LCM. Here, 6 * x = 2 * 24. Solving for x gives 48 / 6 = 8. Therefore, the other number is 8.

A perfect number is equal to the sum of its proper divisors. The first four perfect numbers are 6, 28, 496, and 8128. 8128 is the fourth perfect number, discovered by ancient Greek mathematicians.

An Achilles number is a number that is powerful (each prime factor appears at least squared) but not a perfect power. 108 = 2^2 * 3^3. It is powerful because 2 and 3 are squared/cubed, but it is not a perfect power (like 64 or 81).

Powerful numbers are positive integers where every prime factor appears with at least exponent 2 (they can be expressed as m^2 * n^3). 729 = 3^6 (all prime factors have exponent >=2) and 512 = 2^9 are powerful numbers. 423 = 3 x 141 and 372 = 2^2 x 3 x 31 are not powerful because they have prime factors (3 and 31) with exponent 1.

Square-free numbers are integers not divisible by any perfect square other than 1. 14 = 2 x 7 and 19 (prime) are square-free as they have no repeated prime factors. 12 = 2^2 x 3 and 16 = 2^4 are not square-free because they contain perfect squares (4 and 16 respectively) as divisors.

A practical number is a positive integer n where every positive integer ≤ n can be written as a sum of distinct divisors of n. For 30, divisors are 1,2,3,5,6,10,15,30 and all numbers 1-30 can be formed. For 54, divisors are 1,2,3,6,9,18,27,54 and all numbers 1-54 can be formed. 22 and 44 are not practical numbers - they cannot represent all smaller integers.

A sublime number is a number that has a perfect number of divisors, and the sum of its divisors is also a perfect number. 12 is the smallest sublime number: it has 6 divisors (a perfect number), and their sum is 28 (a perfect number).

A superabundant number is a natural number n such that the ratio σ(n)/n is greater than σ(m)/m for all m < n. 120 is a superabundant number. The other numbers listed do not satisfy this strict maximality condition compared to all smaller integers.

A multiperfect number (or k-perfect number) is a number where the sum of all divisors is an integer multiple of the number itself. These are also known as pluperfect numbers. Pseudoperfect numbers refer to a different property (semiperfect).

A Super Perfect Number is a number n such that σ(σ(n)) = 2n. For n=16: σ(16)=31, σ(31)=32 (216). For n=64: σ(64)=127, σ(127)=128 (264). Both 16 and 64 are super perfect numbers. This is an MC-2 (multiple choice, multiple correct) question.