Question

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

Answer 1

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Please read the explanation.

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.