What is the orthocenter of a triangle with corners at #(7 ,3 )#, #(4 ,8 )#, and (6 ,3 )#?

Answer 1

The orthocenter is #(4, 9/5)#

Determine the equation of the altitude that goes through point #(4,8)# and intersects the line between the points #(7,3) and (6,3)#.

Please notice that the slope of line is 0, therefore, the altitude will be a vertical line:

#x = 4##" [1]"#
This is an unusual situation where the equation of one of the altitudes gives us the x coordinate of the orthocenter, #x = 4#
Determine the equation of the altitude that goes through point #(7,3)# and intersects the line between the points #(4,8) and (6,3)#.
The slope, m, of the line between the points #(4,8) and (6,3)# is:
#m = (3 - 8)/(6 - 4) = -5/2#

The slope, n, of the altitudes will be the slope of a perpendicular line:

#n = -1/m#
#n = 2/5#
Use the slope, #2/5#, and the point #(7,3)# to determine the value of b in the slope-intercept form of the equation of a line, #y = nx + b#
#3 = (2/5)7 + b#
#b = 3 - 14/5#
#b = 1/5#
The equation of the altitude through point #(7,3)# is:
#y = (2/5)x + 1/5##" [2]"#

Substitute the x value from equation [1] into equation [2] to find the y coordinate of the orthocenter:

#y = (2/5)4 + 1/5#
#y = 9/5#
The orthocenter is #(4, 9/5)#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the orthocenter of a triangle, we need to find the point where the three altitudes of the triangle intersect. An altitude is a line segment that extends from one vertex of the triangle perpendicular to the opposite side.

Given the vertices of the triangle as ( (7, 3) ), ( (4, 8) ), and ( (6, 3) ), we can proceed as follows:

  1. Find the slopes of the lines passing through each pair of vertices.
  2. Find the equations of the perpendicular bisectors of each side of the triangle.
  3. Solve the system of equations formed by the perpendicular bisectors to find their point of intersection, which is the orthocenter.

Let's calculate the orthocenter:

  1. Slopes of the sides:

    • Slope of ( (7, 3) ) to ( (4, 8) ): ( m_1 = \frac{8-3}{4-7} = -\frac{5}{3} )
    • Slope of ( (7, 3) ) to ( (6, 3) ): ( m_2 = \frac{3-3}{6-7} = 0 )
    • Slope of ( (4, 8) ) to ( (6, 3) ): ( m_3 = \frac{3-8}{6-4} = -\frac{5}{2} )
  2. Perpendicular bisectors:

    • Midpoint ( M_1 ) of ( (7, 3) ) and ( (4, 8) ): ( (\frac{7+4}{2}, \frac{3+8}{2}) = (5.5, 5.5) ) The perpendicular bisector of ( (7, 3) ) and ( (4, 8) ) is the line perpendicular to the segment ( (7, 3) ) - ( (4, 8) ) passing through ( (5.5, 5.5) ). Its equation is ( x = 5.5 ).
    • Midpoint ( M_2 ) of ( (7, 3) ) and ( (6, 3) ): ( (\frac{7+6}{2}, \frac{3+3}{2}) = (6.5, 3) ) The perpendicular bisector of ( (7, 3) ) and ( (6, 3) ) is the line perpendicular to the segment ( (7, 3) ) - ( (6, 3) ) passing through ( (6.5, 3) ). Its equation is ( y = 3 ).
    • Midpoint ( M_3 ) of ( (4, 8) ) and ( (6, 3) ): ( (\frac{4+6}{2}, \frac{8+3}{2}) = (5, 5.5) ) The perpendicular bisector of ( (4, 8) ) and ( (6, 3) ) is the line perpendicular to the segment ( (4, 8) ) - ( (6, 3) ) passing through ( (5, 5.5) ). Its equation is ( x = 5 ).
  3. The orthocenter is the intersection of the perpendicular bisectors. Since the equations of two of the bisectors are vertical and horizontal lines, the intersection point must lie on both. Thus, the orthocenter is ( (5.5, 3) ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7