The most common shapes in geometry-related problems are lines, triangles, rectangles, squares, circles, parallelograms, rhombus, and trapezoids. Sometimes it involves common solids like spheres, cylinders, rectangular parallelepiped, and cubes.

**Example**

An isosceles triangle has an altitude equal to 12 inches and a perimeter equal to 36 inches. Find the length of the base and the length of the equal sides.

**Solution**

$x + 2y = 36$

$x = 36 - 2y$

$\left( \dfrac{x}{2} \right)^2 + 12^2 = y^2$

$\dfrac{x^2}{4} + 144 = y^2$

$x^2 + 576 = 4y^2$

$(36 - 2y)^2 + 576 = 4y^2$

$(1296 - 144y + 4y^2) + 576 = 4y^2$

$144y = 1872$

$y = 13$

$x = 36 - 2(13) = 10$

Length of base = 10 inches *answer*

Length of equal sides = 13 inches *answer*