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

$x = 36 - 2y$

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

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

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

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

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

$144y = 1872$

$y = 13$

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

Length of base = 10 inches answer Length of equal sides = 13 inches answer

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