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IM Geo.6.17 Lesson: Lines in Triangles

Author:GeoGebra Classroom Activities, IM Algebra 1, Geometry, Algebra 2

If you have tracing paper, draw a triangle on it. Fold the altitude from each vertex. If not, observe the applet below.

Triangle ABC is graphed.

Find the slope of each side of the triangle.

Find the slope of each altitude of the triangle.

  • Sketch the altitudes.
  • Label the point of intersectionĀ .
Write equations for all 3 altitudes.

Use the equations to find the coordinates ofĀ Ā and verify algebraically that the altitudes all intersect atĀ .

Any triangleĀ Ā can be translated, rotated, and dilated so that theĀ imageĀ Ā lies on the origin,Ā Ā lies on the pointĀ , andĀ Ā has positionĀ . Use this as a starting point to prove thatĀ the altitudes of all triangles all meet at the same point.

Triangle ABC is graphed.

Find the midpoint of each side of the triangle.

  • Sketch the perpendicular bisectors, using the Midpoint and Perpendicular Line tools.
  • Label the intersection pointĀ .
Write equations for all 3 perpendicular bisectors.

Use the equations to find the coordinates ofĀ Ā and verify algebraically that the perpendicular bisectors all intersect atĀ .

Consider triangle ABC from an earlier activity.

What is the distance fromĀ Ā toĀ , the intersection point of the perpendicular bisectors of the triangleā€™s sides? Round to the nearest tenth.

Write the equation of a circle with centerĀ Ā and radiusĀ .

  • Construct the circle.
What do you notice?

Verify your hypothesis algebraically.

Consider triangle ABC from earlier activities.

  • Plot pointĀ , the intersection point of the altitudes.
  • Plot pointĀ , the intersection point of the perpendicular bisectors.
  • Find the point where the 3 medians of the triangle intersect. Plot this point and label itĀ .
What seems to be true about pointsĀ , ,Ā andĀ ?

Prove that your observation is true.

A tessellation covers the entire plane with shapes that do not overlap or leave gaps.

Tile the plane with congruent rectangles: Draw the rectangles on your grid.

Write the equations for lines that outline 1 rectangle.

Tile the plane with congruent right triangles: Draw the right triangles on your grid.

Write the equations for lines that outline 1 right triangle.

Tile the plane with any other shapes: Draw the shapes on your grid.

Write the equations for lines that outline 1 of the shapes.

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