What is the orthocenter of a triangle with corners at #(9 , 5 )#, #(3 , 8 )#, and #(5 ,6 )#?

Question

What is the orthocenter of a triangle with corners at #(9 , 5  )#, #(3 , 8   )#, and #(5       ,6   )#?

Autumn Armstrong · Accepted Answer

Steps: (1) find the slopes of 2 sides, (2) find the slopes of the lines perpendicular to those sides, (3) find the equations of the lines with those slopes that pass through the opposite vertices, (4) find the point where those lines intersect, which is the orthocenter, in this case #(6.67, 2.67)#.

The slopes (gradients) of two of a triangle's sides and the equations of the lines perpendicular to those sides are what we need to find the orthocenter of the triangle. The equations of the lines perpendicular to the sides that pass through the opposite angle can be found using those slopes in addition to the coordinates of the point opposite the relevant side; these lines are known as the "altitudes" for the sides. The orthocenter is the point where the altitudes of two sides cross; the altitude of the third side would also pass through this point. To make it simpler to refer to our points, let's label them: Point A = #(9, 5)# Point B = #(3, 8)# Point C = #(5, 6)# Use the formula to determine the slope: #m = (y_2-y_1)/(x_2-x_1)# #m_(AB) = (8-5)/(9-3)=3/6=1/2# #m_(BC) = (6-8)/(5-3)=(-2)/2=-1# We don't want these slopes, though, but the slopes of the lines perpendicular (at right angles) to them. The line perpendicular to a line with slope #m# has slope #-1/m#, so the line perpendicular to #AB# has slope #-2# and the line perpendicular to #BC# has slope #1#. By entering the coordinates of Point C (opposite AB) and Point A (opposite BC) into the equation, we can now find the equations for their respective altitudes. #y=mx+c# The altitude at Point C is: #6 = -2(5)+c# which gives #c=6+10=16# therefore #y = -2x + 16# In a similar vein, Point A: #5=1(9)+c# which gives #c=5-9=-4# so the equation is: #y=x-4# We just need to locate the intersection of these two lines, which we can equate to each other to find the orthocenter: #-2x+16=x-4# Rearranging, #3x=20 to x~~6.67# Substitute into either equation to find the #y# value, which is #2.67#. Therefore the orthocenter is the point #(6.67, 2.67)#.

Caroline Aucoin · Answer

undefined To find the orthocenter of a triangle, you first need to determine the perpendicular bisectors of each side of the triangle. Then, the intersection point of these perpendicular bisectors is the orthocenter.

For the given triangle with vertices at (9, 5), (3, 8), and (5, 6), you can calculate the slopes of the sides, find the midpoints of the sides, and then determine the perpendicular bisectors' equations. Finally, find the intersection point of these bisectors to get the orthocenter.

Alternatively, you can use the formula for the orthocenter, which involves the slopes of the sides and the vertices of the triangle. Once you have the slopes and the vertices, you can substitute them into the formula to find the orthocenter.