Let the length of each equal side be x. The perimeter is given by 2x + 8 = 24. Solving for x gives 2x = 16, so x = 8 cm.

The perimeter is the sum of all sides. Therefore, base = 24 - (10 + 10) = 4 cm.

Perimeter = DE + EF + DF. Therefore, DF = 36 - (10 + 12) = 14 cm.

T = 2π√(r³/GM), thus T ∝ √r.

A pH greater than 7 is considered alkaline.

A pH of 8 indicates alkaline soil.

pOH = 14 - pH = 3, [OH-] = 10^-3 M.

pH = 3 means [H+] = 10^-3 M = 0.001 M.

The concentration of hydrogen ions [H+] is 10^(-pH) = 10^(-3) = 0.001 M.

The photoelectric effect occurs only when the frequency of the incident light is above the threshold frequency.

The paraphrase captures the essence of the original phrase, emphasizing human fallibility.

The paraphrase captures the essence of the original phrase, emphasizing that making mistakes is a common human characteristic.

The rephrased statement captures the essence of the original phrase.

To joke is a synonym for teasing someone.

Being indecisive is synonymous with remaining neutral, as it implies not making a decision.

For a weak acid, pH ≈ pKa for a 0.1 M solution, so pH ≈ 4.76.

Substituting x = 4 gives 3(4) - 4y = 12 => 12 - 4y = 12 => y = 0.

Substituting x = 0: 4(0) - 5y = 20 => -5y = 20 => y = -4.

Slope = (4-2)/(3-1) = 1, and it remains the same for other points.

Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 0.

By Vieta's formulas, the sum of the roots (a + b + c) of the polynomial x^3 - 6x^2 + 11x - 6 is equal to the coefficient of x^2 with the opposite sign, which is 6.

Evaluating f(1) gives 1^3 - 3(1^2) + 4 = 1 - 3 + 4 = 2.

The polynomial can be factored as (x - 1)(x - 2)(x - 3), so x - 1 is one of its factors.

Substituting x = 1 into the polynomial gives f(1) = 1 - 4 + 6 - 4 + 1 = 0.

The polynomial can be rewritten in a form that shows symmetry about the line x = 1.

A polynomial has a double root when its discriminant is equal to zero.

Using Vieta's formulas, the sum of the roots (-2 + 3) = 1, which means b = -1.

Using Vieta's formulas, the sum of the roots (3 + (-2)) = 1, hence b = -1.

According to Vieta's formulas, the sum of the roots r1 + r2 of the polynomial x^2 - 5x + 6 is equal to the coefficient of x (which is -(-5)) = 5.

Factoring the polynomial gives (x - 2)(x - 3), so the roots are 2 and 3.