What is the orthocenter of a triangle with corners at #(2 ,7 )#, #(1 ,1 )#, and (3 ,2 )#?

Question

What is the orthocenter of a triangle with corners at #(2  ,7   )#, #(1   ,1   )#, and (3    ,2   )#?

Jayden Jackson · Accepted Answer

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Please read the explanation.

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The altitude of a triangle is a perpendicular line segment from the vertex of the triangle to the opposite side.

The Orthocenter of a triangle is the intersection of the three altitudes of a triangle.

#color(green)("Step 1"#

Construct the triangle #ABC# with

Vertices #A(2, 7), B(1,1) and C(3,2)#

Observe that #/_ACB = 105.255^@#.

This angle is greater than #90^@#, hence ABC is an Obtuse triangle.

If the triangle is an obtuse triangle, the Orthocenter lies outside the triangle.

#color(green)("Step 2"#

Construct altitudes through the vertices of the triangle as shown below:

All the three altitudes meet at a point referred to as the Orthocenter.

Since the triangle is obtuse, the orthocenter lies outside the triangle.

#color(green)("Step 3"#

Observe that the Orthocenter has #(4.636, 1.727)# as its coordinates.

Hope it helps.

Autumn Armstrong · Answer

undefined To find the orthocenter of a triangle, you need to find the point where the altitudes of the triangle intersect. The altitudes are perpendicular lines drawn from each vertex of the triangle to the opposite side.

Given the coordinates of the triangle's vertices:
A(2, 7), B(1, 1), and C(3, 2),

First, find the slopes of the lines containing each side of the triangle.
Then, use the perpendicular slope relationship to find the slopes of the altitudes.
Next, find the equations of the lines containing the altitudes.
Finally, solve the system of equations to find the point of intersection, which is the orthocenter.