Question

A triangle has vertices at (3, 3), (6, 15), and (3, -3). What are the coordinates of the orthocenter?

A. (-69, 15)
B. (3.5, 3)
C. (4.2)
D. (9, 0)

Answer 1

To find the orthocenter of a triangle with given coordinates, we need to find the equations of the altitudes and solve for their intersection.

Step-by-step explanation:

The orthocenter of a triangle is the point where the three altitudes of the triangle intersect. To find the orthocenter, we first need to find the equations of the altitudes and then solve for their intersection.

The equation of a line passing through two points (x1, y1) and (x2, y2) is given by y - y1 = m(x - x1), where m is the slope of the line. The slope of the line perpendicular to a line with slope m is -1/m.

Using the given coordinates, we can find the slopes of the sides of the triangle and the slopes of the altitudes. Then, we can find the equations of the altitudes and solve them to find the coordinates of the orthocenter.

After performing the calculations, the coordinates of the orthocenter are (4.2, 3).

Answer 2

The answer would be A) (-69, 15)

Solution:

Finding 1st and 2nd

m = (y2 - y1)/(x2 - x1)

m = (15 - 3 )/(6 - 3)

m = 12/3

y = 4x + b

3 = 3(3)+b

3=9+b

-9 -9

-6=b

Finding 3rd

repeat the above process for 1 and 2, so now use 2 and 3 in the equation.

m = -9/3

m=-3

y=-3x+b

15=-3(6)+b

b=-3

Step 2