A triangle is isosceles if it has two sides of equal length....

A triangle is isosceles if it has two sides of equal length. Suppose that an isosceles triangle has two sides of length $a$ and that the third side is the base, of length $b$. It is a fact that the altitude of such a triangle cuts the base in half. Show that the area of the triangle is given by $$ \frac{b}{2} \sqrt{a^2-\frac{b^2}{4}} . $$

Key Concepts

Pythagorean Theorem

The Pythagorean theorem relates the sides of a right-angled triangle, stating that the square of the hypotenuse equals the sum of the squares of the other two sides. In solving for the altitude in an isosceles triangle, one can apply this theorem to one of the resulting right triangles, connecting the known side lengths to the unknown height.

Isosceles Triangle Properties

An isosceles triangle has at least two sides of equal length, which creates a line of symmetry through the vertex opposite the base. This symmetry ensures that when an altitude is drawn from the vertex to the base, it also bisects the base, forming two congruent right-angled triangles. This property is essential in simplifying geometric calculations involving the triangle.

Altitude in Triangles

The altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. In an isosceles triangle, the altitude from the apex not only provides the height necessary for calculating the area but also divides the base into two equal parts, allowing the application of right triangle principles.

Area of a Triangle

The area of a triangle is calculated as one-half the product of its base and its altitude. After determining the altitude using the Pythagorean theorem in the context of an isosceles triangle, this formula is used to derive an expression for the area. This concept of area calculation is fundamental in geometry.

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