* Median . * A median A_1M_1 of a triangle DeltaA_1A_2A_3 is the Cevian from one of its vertices A_1 to the midpoint M_1 of the opposite side. The three medians of any triangle are concurrent (Casey 1888, p. 3), meeting in the triangle centroid (Durell 1928) G, which has trilinear coordinates 1/a:1/b:1/c. In addition, the medians of a triangle divide one another in the ratio 2:1 (Casey 1888, p. 3). A median also bisects the area of a triangle. Let m_i denote the length of the median of the ith side a_i. Then m_1^2 = 1/4(2a_2^2+2a_3^2-a_1^2) (1) m_1^2+m_2^2+m_3^2 = 3/4(a_1^2+a_2^2+a_3^2) (2) (Casey 1888, p. 23; Johnson 1929, p. 68). The area of a triangle can be expressed in terms of the medians by A=4/3sqrt(s_m(s_m-m_1)(s_m-m_2)(s_m-m_3)), (3) where s_m=1/2(m_1+m_2+m_3). * Angle Bisector . * An angle bisector of an angle of a triangle divides the opposite side in two segments that are proportional to the other two sides of the triangle. By the Angle Bisector Theorem, Proof: Extend to meet at point E. By the Side-Splitter Theorem, ---------(1) The angles are corresponding angles. So, Since is a angle bisector of the angle By the Alternate Interior Angle Theorem, Therefore, by transitive property, Since the angles are congruent, the triangle is an isosceles triangle with AE = AB. Replacing AE by AB in equation (1), Example : Find the value of x. By Triangle-Angle-Bisector Theorem, Substitute. Cross multiply. 5x = 42 Divide both sides by 5. The value of x is 8.4. * Vertices * (a). Use the two-point form of an equation of a line: where and are the coordinates of points A and C. (b). Step 1. Use the portion of the two-point form to determine the slope of the line containing segment AB. Step 2: Calculate the negative reciprocal of the slope determined in Step 1 because: Step 3: Use the point-slope form of an equation of a line: where are the coordinates of point C and is the slope calculated in (b) Step 2. (c) Step 1: Use the mid-point formulas: and where and are the coordinates of points A and C to calculate the midpoint of segment AC. Use the slope calculated in (b) Step 2 and the midpoint calculated in (c) Step 1 with the point-slope form to derive the equation of the perpendicular bi-sector of AC. (d) Step 1: Use the distance formula 3 times: where , , and are the coordinates of points A, B, and C . Step 2: Sum the three results. John My calculator said it, I believe it, that settles it * Equilateral * A triangle which has all three of its sides equal in length. Try this Drag the orange dots on each vertex to reshape the triangle. Notice it always remains an equilateral triangle. The sides AB, BC and AC always remain equal in length An equilateral triangle is one in which all three sides are congruent (same length). Because it also has the property that all three interior angles are equal, it really the same thing as an equiangular triangle. See Equiangular triangles. An equilateral triangle is simply a specific case of a regular polygon, in this case with 3 sides. All the facts and properties described for regular polygons apply to an equilateral triangle. See Regular Polygons Properties All three angles of an equilateral triangle are always 60°. In the figure above, the angles ∠ABC, ∠CAB and ∠ACB are always the same. Since the angles are the same and the internal angles of any triangle always add to 180°, each is 60°. The area of an equilateral triangle can be calculated in the usual way, but in this special case of an equilateral triangle, it is also given by the formula: area of equilateral triangle formula Calculator where S is the length of any one side. See Area of an equilateral triangle. With an equilateral triangle, the radius of the incircle is exactly half the radius of the circumcircle.
Posted on: Sat, 25 Jan 2014 23:35:39 +0000
Trending Topics
Recently Viewed Topics
© 2015