Given the binomials (x − 2), (x − 1), (x + 2), and (x − 4), which one is a factor of f(x) = x^3 + 7x^2 + 14x + 8? Could you also
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Answer:
- (1,3) is inside the triangle
Step-by-step explanation:
Orthocenter is the intersection of altitudes.
We'll calculate the slopes of the two sides and their altitudes ad find the intersection.
<h3>Side QR</h3>- m = (3 - 5)/(4 - (-1)) = -2/5
<u>Perpendicular slope:</u>
- -1/m = 5/2
<u>Perpendicular line passes through S(-1, -2):</u>
- y - (-2) = 5/2(x - (-1)) ⇒ y = 5/2x + 1/2
- m = (-2 - 3)/(-1 -4) = -5/-5 = 1
<u>Perpendicular slope:</u>
- -1/m = -1/1 = -1
<u>Perpendicular line passes through Q(-1, 5):</u>
- y - 5 = -(x - (-1)) ⇒ y = -x + 4
The intersection of the two lines is the orthocenter.
<u>Solve the system of equations to get the coordinates of the orthocenter:</u>
- 5/2x + 1/2 = x + 4
- 5x + 1 = -2x + 8
- 7x = 7
- x = 1
<u>Find y-coordinate:</u>
- y = -1 + 4 = 3
The orthocenter is (1, 3)
If we plot the points, we'll see it is inside the triangle
