คณิตพื้นฐาน

A · The hypotenuse

The hypotenuse is the side opposite the right angle and is always the longest side in a right-angled triangle.

C · $a^2 + b^2 = c^2$

The Pythagorean theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides: $c^2 = a^2 + b^2$.

A · 5

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

D · $\sqrt{2}$

An irrational number cannot be expressed as a fraction of two integers. $\sqrt{2}$ is a classic example of an irrational number.

C · 9

The square root of 81 is 9 because $9^2 = 81$. Note: The principal square root is 9, but both 9 and -9 are square roots.

C · -3

The cube root of -27 is -3 because $(-3)^3 = -27$.

C · Base area × height

The volume of a prism is calculated by multiplying the area of its base by its height.

C · $0.333...$

$0.333...$ is a repeating decimal, which can be written as the fraction $\frac{1}{3}$, so it is a rational number.

B · 12

The square root of 144 is 12 because $12^2 = 144$.

D · 60 cm³

Volume = length × width × height = $5 \times 3 \times 4 = 60$ cm³.

C · 7 and -7

The equation $x^2 = 49$ has two solutions: $x = 7$ and $x = -7$, because both $7^2$ and $(-7)^2$ equal 49.

B · False

0 is a rational number because it can be written as $\frac{0}{1}$, where both numerator and denominator are integers and the denominator is not zero.

A · True

The hypotenuse is opposite the right angle, which is the largest angle in a right triangle, so the hypotenuse is always the longest side.

D · $2\pi rh + 2\pi r^2$

The surface area of a cylinder includes the lateral surface area ($2\pi rh$) plus the area of the two circular bases ($2\pi r^2$).

A · Right triangle

Check $c^2 = a^2 + b^2$: $10^2 = 6^2 + 8^2$ gives $100 = 36 + 64 = 100$, so it is a right triangle.

A · The bases are always circles

A prism has polygonal bases, not circular bases. Circular bases are characteristic of cylinders.

A · 96 cm²

A cube has 6 faces, each with area $4^2 = 16$ cm². Total surface area = $6 \times 16 = 96$ cm².