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Altitudes

Height of a triangle or the line segment from a vertex and perpendicular to the opposite side.

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Altitudes

What if you were given one or more of a triangle's angle measures? How would you determine where the triangle's altitude would be found? After completing this Concept, you'll be able to answer this type of question.

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CK-12 Altitudes

James Sousa: Altitudes of a Triangle

Guidance

In a triangle, a line segment from a vertex and perpendicular to the opposite side is called an altitude. It is also called the height of a triangle. The red lines below are all altitudes.

When a triangle is a right triangle, the altitude, or height, is the leg. If the triangle is obtuse, then the altitude will be outside of the triangle. If the triangle is acute, then the altitude will be inside the triangle.

Example A

Which line segment is the altitude of \begin{align*}\triangle ABC\end{align*}?

Solution: In a right triangle, the altitude, or the height, is the leg. If we rotate the triangle so that the right angle is in the lower left corner, we see that leg \begin{align*}BC\end{align*} is the altitude.

Example B

A triangle has angles that measure \begin{align*}55^\circ, 60^\circ,\end{align*} and \begin{align*}65^\circ\end{align*}. Where will the altitude be found?

Solution: Because all of the angle measures are less than \begin{align*}90^\circ\end{align*}, the triangle is an acute triangle. The altitude of any acute triangle is inside the triangle.

Example C

A triangle has an angle that measures \begin{align*}95^\circ\end{align*}. Where will the altitude be found?

Solution: Because \begin{align*}95^\circ > 90^\circ\end{align*}, the triangle is an obtuse triangle. The altitude of any obtuse triangle is outside the triangle.

CK-12 Altitudes

Guided Practice

1. True or false: The altitudes of an obtuse triangle are inside the triangle.

2. Draw the altitude for the triangle shown.

Solution: The triangle is an acute triangle, so the altitude is inside the triangle as shown below so that it is perpendicular to the base.

3. Draw the altitude for the triangle shown.

Solution: The triangle is a right triangle, so the altitude is already drawn. The altitude is \begin{align*}\overline{XZ}\end{align*}.

Answers

1. Every triangle has three altitudes. For an obtuse triangle, at least one of the altitudes will be outside of the triangle, as shown in the picture at the beginning of this concept.

2.

3.

Practice

Given the following triangles, tell whether the altitude is inside the triangle, outside the triangle, or at the leg of the triangle.

  1. \begin{align*}\triangle JKL\end{align*} is an equiangular triangle.
  2. \begin{align*}\triangle MNO\end{align*} is a triangle in which two the angles measure \begin{align*}30^\circ\end{align*} and \begin{align*}60^\circ\end{align*}.
  3. \begin{align*}\triangle PQR\end{align*} is an isosceles triangle in which two of the angles measure \begin{align*}25^\circ\end{align*}.
  4. \begin{align*}\triangle STU\end{align*} is an isosceles triangle in which two angles measures \begin{align*}45^\circ\end{align*}.

Given the following triangles, which line segment is the altitude?

Vocabulary

altitude

altitude

An altitude of a triangle is a line segment from a vertex and is perpendicular to the opposite side. It is also called the height of a triangle.

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