Drawing altitudes for an obtuse angle triangle often requires extending the sides because two of the three altitudes fall outside the triangle.
Understanding Altitudes in Obtuse Triangles
An altitude of a triangle is a perpendicular line segment from a vertex to the opposite side, or to the line containing the opposite side.
In an obtuse angle triangle, one angle is greater than 90 degrees. This affects where the altitudes land:
- The altitude from the obtuse angle vertex falls inside the triangle.
- The altitudes from the two acute angle vertices fall outside the triangle, requiring the extension of the opposite side.
The point where all three altitudes (or their extensions) intersect is called the orthocenter. For an obtuse triangle, the orthocenter is always outside the triangle.
Drawing Altitudes from Acute Angles (Outside the Triangle)
Drawing an altitude from an acute angle vertex requires extending the opposite side. Here is a step-by-step process, incorporating techniques like those described in geometric constructions:
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Extend the Base: Choose one side opposite an acute angle (e.g., side PQ in triangle PQR). Extend this side in both directions beyond the triangle's vertices using a ruler or straightedge. This extended line forms the base for the altitude construction.
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Set the Compass: Set the compass point on the opposite vertex (let's call this vertex R, as in the reference). Set the compass width wide enough so that when you draw an arc, it will cross the extended line PQ in two distinct places.
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Draw Arcs Across the Line: With the compass point fixed at vertex R, make two arcs that intersect the extended line PQ. Label the points where the arcs cross the line as A and B. These two points are now equidistant from vertex R on the extended line.
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Find the Perpendicular Point: Now, place the compass point on A and set the width to more than half the distance between A and B. Draw an arc. Without changing the compass width, place the compass point on B and draw another arc. These two arcs should overlap, creating a new point (label it C, as in the reference). Point C, along with R, defines the perpendicular line.
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Draw the Altitude: Draw a straight line segment from the vertex R, passing through point C, and extending to the extended line PQ. The point where this line segment intersects the extended line PQ is the foot of the altitude. The altitude itself is the segment from vertex R to this intersection point on the extended line.
You must repeat this process for the other acute angle vertex to draw the second altitude that falls outside the triangle.
Drawing the Altitude from the Obtuse Angle (Inside the Triangle)
Drawing the altitude from the obtuse angle vertex is similar to drawing an altitude in an acute or right triangle because it falls inside the triangle.
- From the vertex of the obtuse angle, drop a perpendicular line straight down to the opposite side. This can often be done with a ruler and a set square or by using compass construction techniques (similar to drawing a perpendicular from a point to a line segment where the point is not on the segment).
The point where this altitude meets the opposite side is its foot.
Summary
Drawing altitudes in an obtuse triangle involves two different approaches depending on the vertex:
| Altitude From | Location of Altitude | Construction Method |
|---|---|---|
| Acute Angle (1) | Outside the triangle | Extend the opposite side; drop a perpendicular from the vertex to the extended side. |
| Acute Angle (2) | Outside the triangle | Extend the opposite side; drop a perpendicular from the vertex to the extended side. |
| Obtuse Angle (1) | Inside the triangle | Drop a perpendicular from the vertex directly to the opposite side. |
All three altitudes (two outside, one inside) or their extensions will meet at a single point, the orthocenter, which is located outside the obtuse triangle.
