Given triangle ABC with coordinates A(−2, 7), B(−2, 4), and C(−4, 3), and its image A′B′C′ with A′(2, 3), B′(−1, 3), and C′(−2, 1), find the line of reflection. The line of reflection is at y =

y = x + 5

Step-by-step explanation:

given an obtuse scalene triangle (∆ABC) with vertices A(-2,7), B(-2,4), and C(-4,3) (representing the preimage of the reflection). And congruent triangle (∆A'B'C') with vertices A'(2,3), B'(-1,3), and C'(-2,1) (representing the image of the reflection / triangle after the reflection).

The easiest way to look at this is by taking the inverse of each point on the preimage (swapping y and x coordinates) and the adding 5 to the y and subtracting 5 from the x to get the image.

You can test this, and you will see that this will match with vertices of the image.

- 2 + 10x³ - 21x² + 19x - 4

step-by-step explanation:

each term in the second factor is multiplied by each term in the first factor

x²(- 2x² + 4x - 1) - 3x(- 2x² + 4x - 1) + 4(- 2x² + 4x - 1) ← distribute

= - 2 + 4x³ - x² + 6x³ - 12x² + 3x -8x² + 16x - 4

collect like terms

= - 2 + 10x³ - 21x² + 19x - 4 ← in standard form

step-by-step explanation:

assume that you have 3 different side lengths.   apply the pythagorean theorem here:   square each of the two shortest sides; call them a and b, so that their squares are a² and b².   also square the longest side, calling the result c².

now, if a² + b² = c² is true, the triangle is a right triangle; otherwise it is not a right triangle.