Question

What are the coordinates of the orthocenter of △xyz with vertices at x(−3, 3), y(1, 3), and z(−3, 0)? Enter your answer in the boxes. ( , )

a) (1, 3)
b) (-3, 0)
c) (-3, 3)
d) Insufficient information provided

Answer 1

The coordinates of the orthocenter of triangle XYZ with vertices X(-3, 3), Y(1, 3), and Z(-3, 0) are (-5/3, 0).

Step-by-step explanation:

The orthocenter of a triangle is the point of intersection of the altitudes of the triangle. To find the orthocenter of triangle XYZ with vertices X(-3, 3), Y(1, 3), and Z(-3, 0), we need to find the equations of the altitudes and solve for their point of intersection.

Step 1: Find the slopes of the sides XY, YZ, and XZ.

Slope of XY = (3 - 3) / (-3 - 1) = 0

Slope of YZ = (3 - 0) / (1 - (-3)) = 3/4

Slope of XZ = (3 - 0) / (-3 - (-3)) = undefined

Step 2: Find the equations of the altitudes using the slopes and the midpoint of each side.

Equation of altitude from X to YZ: y - 3 = 0(x + 3) => y = 0

Equation of altitude from Y to XZ: y - 3 = (3/4)(x - 1) => y = (3/4)x + 5/4

Equation of altitude from Z to XY: x - (-3) = 0(y - 0) => x = -3

Step 3: Solve the system of equations to find the point of intersection of the altitudes.

Setting the two altitudes' equations equal to each other, we get (3/4)x + 5/4 = 0, which simplifies to x = -5/3.

Substituting the value of x into one of the altitude equations, we find y = 0.

Therefore, the coordinates of the orthocenter of triangle XYZ are (-5/3, 0).