Calculating Areas of Special Triangles Where Is Necessary to Find the Height, and/or the Base.

The area for a triangle is one half the product of the base by the height; nevertheless, sometimes this is not possible to apply right away. In those cases we need to use special right triangles (30°-60°-90°, or 45°-45°-90°), and/or Pythagorean Theorem, or trigonometry; to find the base, and/or the height for a particular problem.

We are aware that these are topics that present sometimes difficulties for some students, that is why we strive to develop the problem solving process in a detailed step by step mode, and without skipping, or implying that the student should know it. Whether you are a struggling student, or an advanced one; you will find the lesson to be doable, interesting, and challenging fun!

Lesson's Content

Lesson's Glossary

Altitude
Height

Altitude of a triangle (height)
The perpendicular segment from a vertex to the line containing the opposite side of a triangle.

Angle
Geometry shape formed by two rays (initial and ending sides of the angle) that share a common endpoint called the vertex. You may name an angle using the vertex, or a point in each ray and the vertex label in the center.

Angle bisector
A ray that is in the interior of an angle and forms two equal angles with the sides of that angle. Area
The amount of space taken up in a plane by a figure.

Area
The amount of space taken up in a plane by a figure.

Base
The side of an isosceles triangle whose endpoints are the vertices of the base angles.

Base angle
The angle opposite one of the equilateral sides in an isosceles triangle.

Cosine
Ratio determined by the hypotenuse in a right triangle and a side adjacent to a reference angle.

Equilateral triangle
A triangle whose sides are equal in length.

Isosceles triangle
A triangle with two sides of equal length.

Median
The segment connecting the vertex of an angle in a triangle to the midpoint of the side opposite it.

Obtuse angle
An angle whose measure is greater than 90 but less than 180 degrees

Obtuse triangle
A triangle with one acute angle.

Perpendicular bisector
The bisector of a segment perpendicular to it.

Polygon
It is a closed plane figure with a least three straight segments as sides.

Pythagorean theorem a2 + b2 = c2.
The Pythagorean theorem states that if you have a right triangle, then the square built on the hypotenuse is equal to the sum of the squares built on the other two sides.

Right triangle
A triangle that has a 90 degree angle.

Sine
Ratio of the length of the opposite side to a reference angle in a right triangle and the hypotenuse. Opposite/hypotenuse.

Scalene triangle
A triangle with no equilateral sides.

Triangle
A polygon with three sides.

Tangent
The ratio of the length of the opposite side of a reference angle to the adjacent side to the same angle. Opposite over adjacent.

Interactive Geometric Applets: Relevant Theorems.

Finding the area for special triangles sometimes will require that you use the side

of the triangle with the base, and apply the Pythagorean Theorem. This applet

shows in a dynamic way how the theorem works.

Hold any vertex and move it around to see how the values in the

equation are updated, and how they comply with the theorem.

There are problems in which to find the area of the triangle, you can't use the Pythagorean Theorem,

or special right triangles; you are left with trigonometry as your only option. You may drag the

vertex in the triangle below to see how the trigonometric ratios are updated, and their

relationship with their inverses.

Part of solving the areas for special triangles consist in understanding basic definitions

for special segments in triangles. These are altitudes, angle bisectors, medians, and perpendicular

bisectors.

Practice in this applet dragging any of the vertices in the triangles, and read the definitions;

check how they comply.

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