Question

50 points! Tell whether the orthocenter is inside, on, or outside the triangle. Then find the coordinates of the orthocenter.

Q(-1, 5), R(4, 3), S(-1, -2)

Answer 1

• (1,3) is inside the triangle

Explanation:

Orthocenter is the intersection of altitudes.

We'll calculate the slopes of the two sides and their altitudes ad find the intersection.

Side QR

• m = (3 - 5)/(4 - (-1)) = -2/5

Perpendicular slope:

• -1/m = 5/2

Perpendicular line passes through S(-1, -2):

• y - (-2) = 5/2(x - (-1)) ⇒ y = 5/2x + 1/2

Side RS

• m = (-2 - 3)/(-1 -4) = -5/-5 = 1

Perpendicular slope:

• -1/m = -1/1 = -1

Perpendicular line passes through Q(-1, 5):

• y - 5 = -(x - (-1)) ⇒ y = -x + 4

The intersection of the two lines is the orthocenter.

Solve the system of equations to get the coordinates of the orthocenter:

• 5/2x + 1/2 = x + 4
• 5x + 1 = -2x + 8
• 7x = 7
• x = 1

Find y-coordinate:

• y = -1 + 4 = 3

The orthocenter is (1, 3)

If we plot the points, we'll see it is inside the triangle