Geometry - Triangle Problems and Solutions
Triangle, solved problems, examples
Example:   A circle with the center at the center of an equilateral triangle intersects all three sides of the triangle to three equal parts each. Find the area of the parts of the triangle outside the circle if the side of the triangle a = Ö6.
Solution:  Given,   a = Ö6.   A = ?
Example:   Lengths of sides of a right triangle are terms of an arithmetic progression with the difference 2, find the radius of incircle.
Solution:  Given,  a = xb = x + and  c = x + 4.   r = ?
Example:   A line parallel to the side AB, of a triangle ABC, divides the triangle to two parts of the same areas and intersects the side AC at the point A1 and the side BC at the point B1. If radii of incircles of triangles, ABC and A1B1C are at the same time legs of a right triangle whose hypotenuse length is 4Ö3, find the radius of the incircle of the triangle ABC.
 Solution:
Example:   In a right triangle ABC, with the angle 90° at C, the altitude from vertex C to the hypotenuse AB meets the hypotenuse at the point D. If both radii of incircles of triangles ACD and BCD are equal and measure 2Ö2 cm, find the radius (r) of the incircle of the triangle ABC.
Solution:  Given,  r1 = 2Ö2 cm.   r = ?
Example:   In a right triangle ABC, with the angle 90° at C, the altitude from vertex C to the hypotenuse AB meets the hypotenuse at the point D. If radii of incircles of triangles ACD and BCD are, 1 cm and 2Ö2 cm rspectively, find the radius of the incircle of the triangle ABC.
Solution:  Given,  r1 = 1 cm  and  r2 = 2Ö2 cm.  r = ?
 Since,  DABC ~ DACD ~ DBCD  then,
Example:   Lengths of sides of a triangle are in the proportion 9 : 10 : 17, find the radius of the incircle if the difference of the longest and the shortest side is 16 cm.
Solution:   Given,  a : b : = 9 : 10 : 17 and  c - a = 16 cm.  r = ?
Example:   The side AB of a triangle ABC is 9 cm long. Sides, AC and BC, form on a line l, which is parallel to side AB, the segment 3 cm long. If the distance of the vertex C from the line l is 3 cm, find the distance of the line l from the side AB.
Solution:   As shown triangles are similar following proportion must hold,

 9 : 3 = (3 + x) : 3 3 · (3 + x) = 3 · 9 3 + x = 9 x = 6 cm.
Geometry and use of trigonometry contents - A