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# Part 7

 Theorem 8.9 Corresponding altitudes of similar triangles are proportional to corresponding sides. Theorem 8.10 Corresponding medians of similar triangles are proportional to corresponding sides. Theorem 8.11 The bisector of an angle of a triangle divides the opposite side of the triangle into segments proportional to the other two sides. Theorem 9.1 In a right triangle, the altitude to the hypotenuse forms two similar right triangles, each of which is also similar to the original triangle. Corollary 1 In a right triangle, the square of the length of the altitude to the hypotenuse equals the product of the lengths of the segments formed on the hypotenuse. Corollary 2 If the altitude is drawn to the hypotenuse of a right triangle, then the square of the length of either leg equals the product of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg. Theorem 9.2 Pythagorean Theorem: In any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. Theorem 9.3 Converse of the Pythagorean Theorem: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. Theorem 9.4 If the square of the longest side of a triangle is greater (less) than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse (acute). Theorem 9.5 In a 45-45-90 triangle, the hypotenuse is N/2- times as long as a leg. Theorem 9.6 In a 30-60-90 triangle, the hypotenuse is twice as long as the leg opposite the 30 angle. The leg opposite the 60 angle is Ö3 times as long as the leg opposite the 30 angle. Theorem 10.1 Distance Formula: The distance d between P1(x1,y1) and P2(x2,y2) is given by the formula d = Ö (X2 - X1)^2 + (Y2 - Y1)^2 Theorem 10.2 Midpoint Formula: Given P1(xl,yl) and P2(X2,Y2), the coordinates (x ... y .. ) of M, the midpoint of PQ, are (x1+x2)/2, (y1+y2)/2. Theorem 10.3 All segments of a non-vertical line have equal slopes. Theorem 10.4 An equation of a line with slope m containing the point P1(xl,yl) is y - y1 = m(x - x1). Theorem 10.5 If a line has slope m and y-intercept b, then an equation of the line is y = mx + b. Theorem 10.6 If two non-vertical lines have the same slope, then they are parallel.

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Note: The concepts in this collection may not be entirely accurate.
They are for reference only.