Find point C on the x-axis so that AC+BC is a minimum. A(−1,7), B(5,−4)Write a rule for the translation of △LMN to △L′M′N′.

The coordinate of the point C is (-10/3, 0)

Step-by-step explanation:

The coordinates of the given points are A(-1, 7), B(5, -4)

Let the coordinate of the point C on the x-axis = C(x, 0)

Therefore, we have;

The square of the distance, d², between two points having coordinates, (x₁, y₁), and (x₂, y₂),  is given by the following formula;

d² = (x₂ - x₁)² + (y₂ - y₁)²

The square of the distance from the point C to A, AC², is given as follows;

AC² = (x - (-1))² + (0 - 7)² = (x + 1)² + 7² = x² + 2·x + 1 + 49 = x² + 2·x + 50

Similarly, the square of the distance from the point C to B, AB², is given as follows;

BC² = (x - (-4))² + (0 - 5)² = (x + 4)² + 5² = x² + 10·x + 25 + 25= x² + 10·x + 50

Therefore;

AC² + BC² = x² + 2·x + 50  + x² + 10·x + 50 = 2·x² + 12·x + 100

(AC + BC)² = AC² + BC² + 2·AB·BC

2·AB·BC = √(4·AB²·BC²) =√(4 × (x² + 2·x + 50) × (x² + 10·x + 50))

2·AB·BC = √(4·x⁴ + 48·x³ + 480·x² + 2400·x + 10,000)

(AC + BC)² = 2·x² + 12·x + 100 + √(4·x⁴ + 48·x³ + 480·x² + 2400·x + 10,000)

AB + BC will be smallest when (AC + BC)² is smallest

Differentiating (AC + BC)² and equating to zero, to find the minimum point, gives;

d(2·x² + 12·x + 100 + √(4·x⁴ + 48·x³ + 480·x² + 2400·x + 10,000))/dx =

Solving using an online application, gives x = -10/3

Therefore, the point C has coordinates C(-10/3, 0).

let's solve your equation step-by-step.

4x+2=2x+12

step 1: subtract 2x from both sides.

4x+2−2x=2x+12−2x

2x+2=12

step 2: subtract 2 from both sides.

2x+2−2=12−2

2x=10

step 3: divide both sides by 2.

2x

2

=

10

2

x=5

step-by-step explanation:

the system of inequalities is

step-by-step explanation:

step 1

find the equation of the dashed line

we have the points

(0,0) and (3,1)

the line passes through the origin, so, is a direct variation

find the slope

the equation of the dashed line is

the solution of the inequality a s the shaded area above the dashed line

therefore

the equation of the inequality a is

step 2

find the equation of the solid line

we have the points

(4,0) and (0,4)

find the slope

the equation of the solid line is

the solution of the inequality b s the shaded area below the solid line

therefore

the equation of the inequality b is

step 3

the system of inequalities is