Use similar triangles to find the length of the altitudes labeled with variables in each triangle below. Similarity: Right triangles, altitudes, and similarity patterns. Redraw triangles and write and solve proportions as needed. Label the segment AD as x, the segment DC as y and the segment BD as z.

(a) Draw the altitude BD from vertex B to the line containing AC. Similarity: Right triangles, altitudes, and using similarity to find unknown values. _, _, _ (f) Summarize what we know about the triangles formed by an altitude from the right angle of a right triangle.

(e) Identify the three triangles by name be sure to name each one in the order of the corresponding parts. (d) Are the triangles similar? Explain how you know.

Label and mark all angles as they are marked in the original diagram. (a) How many triangles do you see in the figure?_ (b) Mark A and C with 2 different marks or colors. In triangle ABC below, BD is the altitude from vertex B to the line containing AC. Similarity: Right triangles, altitudes, and similarity Recall that an altitude of a triangle is a perpendicular line segment from a vertex to the line determined by the (c) Explain how you found the lengths in part (b). (a) Are the triangles at right similar? Explain. Similarity: Right triangles and similarity. LO: I can use similarity to solve problems with altitudes in right triangles. (1) What do you think the word altitude means? (2) Use the word altitude in a sentence. Since both arrangements yield the same triangle, the areas of the square and the rectangle must be identical.DO NOW Geometry Regents Lomac 2014-2015 Date. One such arrangement requires a square of area h 2 to complete it, the other a rectangle of area pq. Based on dissection and rearrangement Дissecting the right triangle along its altitude h yields two similar triangles, which can be augmented and arranged in two alternative ways into a larger right triangle with perpendicular sides of lengths p+h and q+h. ДЄ division by two finally yields the formula of the geometric mean theorem. | C D | | D E | = | A D | | D B | в‡” h 2 = p q. Geometric mean theorem as a special case of the chord theorem: Since the altitude is always smaller or equal to the radius, this yields the inequality. Now the altitude represents the geometric mean and the radius the arithmetic mean of the two numbers. For the numbers p and q one constructs a half circle with diameter p+q. ДЄnother application of provides a geometrical proof of the AMвЂ“GM inequality in the case of two numbers. The method also allows for the construction of square roots (see constructible number), since starting with a rectangle that has a width of 1 the constructed square will have a side length that equals the square root of the rectangle's length. Due to Thales' theorem C and the diameter form a right triangle with the line segment DC as its altitude, hence DC is the side of a square with the area of the rectangle. Then we erect a perpendicular line to the diameter in D that intersects the half circle in C. Now we extend the segment q to its left by p (using arc AE centered on D) and draw a half circle with endpoints A and B with the new segment p+q as its diameter. For such a rectangle with sides p and q we denote its top left vertex with D. The latter version yields a method to square a rectangle with ruler and compass, that is to construct a square of equal area to a given rectangle.

8 1 similarity in right triangles