Question

A right triangle in the first quadrant has the coordinate axes as sides, and the hypotenuse passes through the point (1,8). Find the vertices of the triangle such that the length of the hypotenuse is minimum.

Answer 1

±2, ±3 are potential roots of the polynomial p(x) =
`x^4 − 9x^2 − 4x − 12.`By applying the Rational Root Theorem, testing divisors of the constant term by divisors of the leading coefficient, only ±2 and ±3 satisfy the polynomial equation, resulting in the identified roots.

Step-by-step explanation:

To find potential roots of the polynomial, we can use the Rational Root Theorem. According to this theorem, potential roots are the divisors of the constant term (in this case, 12) divided by the divisors of the leading coefficient (in this case, 1). The divisors of 12 are ±1, ±2, ±3, ±4, ±6, ±12, and the divisors of 1 are ±1. Therefore, the potential rational roots are ±1, ±2, ±3, ±4, ±6, ±12.

Now, we substitute each potential root into the polynomial and check for which values p(x) equals zero. After evaluating, we find that p(2) = 0, p(-2) = 0, p(3) = 0, and p(-3) = 0. Therefore, ±2, ±3 are the roots of the polynomial. The other potential roots (±1, ±4, ±6, ±12) do not satisfy the equation. This is the reason they are not included in the final answer. Thus, the roots of the given polynomial are ±2, ±3.

Answer 2

The vertices of the triangle with minimum hypotenuse are A (0, 0), B (√(65), 0), and C (1, 8).

Consider a right triangle with vertices A (0, 0), B (x, 0), and C (1, 8), where the hypotenuse AC passes through the point (1, 8). Let D be the foot of the perpendicular from C to AB.

Using the Pythagorean theorem, we can express the length of the hypotenuse AC as:

AC^2 = AB^2 + BC^2

Substituting the given values, we get:

(1^2 + 8^2) = x^2 + 0^2

Solving for x, we find:

x = √(65)

To minimize the length of the hypotenuse AC, we need to minimize the distance CD. This distance can be expressed as:

CD = 8 / √(65)

Therefore, the vertices of the triangle with minimum hypotenuse are A (0, 0), B (√(65), 0), and C (1, 8).