7. State and prove the Law of Sines for triangles in Euclidean geometry. 8. Assume Euclidean geometry. Fix a circle and let AB and CD be two chords of the circle that intersect at point P. Prove that AP × PB = CP × PD (one both sides of the equation you are multiplying the lengths) 7. State and prove the Law of Sines for triangles in Euclidean geometry. 8. Assume Euclidean geometry. Fix a circle and let AB and...
Prove in Euclidean Geometry RS^2=RA*RT S T
5. Assume Euclidean geometry. Prove the following: if a trapezoid has congruent legs i.e. the non-parallel sides have the same length), then the angles at the base of the trapezoid are congruent 6.Assume Euclidean geometry. Let ABCD be a trapezoid with ADI BC and with AB-AD. Show that BD bisects angle LABC. 5. Assume Euclidean geometry. Prove the following: if a trapezoid has congruent legs i.e. the non-parallel sides have the same length), then the angles at the base of...
euclidean geometry step by step process 1. (7 pts) Prove that the diagonals of a rectangle are congruent. 2. (18 pts) In the diagram below, prove that M is the midpoint of AC and BD if and only if ABCD is a parallelogram. 3. (9 pts) Use #1 and #2 above to prove that the diagonals of a square cut each other into 4 congruent segments. Use #3 to prove that the diagonals of a square are angle bisectors of...
1. 1. (Absolute Geometry) Assume points A, D, C, B satisfy A-D-C and B is not on the line determined by A, D, C. Prove Internal Angle Sum of triangle ACB is less than or equal to Internal Angle Sum of triangle ADB. (NOTE: This is not Euclidean Geometry. Prove this in Absolute Geometry.)
9. ( 20 points.) In the Cartesian plane model of Euclidean geometry, which of the triples of points (a)-(d) below, if any, are the vertices of a right triangle? (a) (2, 1), (7,0),(5, 7). (b) (102,51), (101, 48), (105,57). (e) (2,1),(4,0),(4,7). (d) (102. - 49), (104,-50). (105,-43). (c) None
Part III (3 pts) For cach of the property statement below, determine which geometry would BEST xhoi given property (choose only one!). Please use A. for Euclidean geometry, B. for hypere geometry, gcometry and D. for Neutral geometry for your identifications Example. A There is a triangle in which the sum of the measures of the interior angles is 180. a. The opposite sides of a parallelogram are congruent. b. Similar triangles may not be congruent. Lines perpendicular to the...
(a) Prove the Chernoff bound: PLY 2ESEe v20 (b) Find the tightest Chernoff bound on PLY2E for fy) (a) Prove the Chernoff bound: PLY 2ESEe v20 (b) Find the tightest Chernoff bound on PLY2E for fy)
(9) Let E R" and let A E L(R"). Define a map f : R" -> R" by f (x) A,)v. Here (is the Euclidean inner product (a) Prove that f is a C1 map and find f'(x) (b) Prove that there exist two that f U V is a bijection on R" neighborhoods of the origin in R", U and V, such (9) Let E R" and let A E L(R"). Define a map f : R" -> R"...
2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove that if the converse of the Alternate Interior Angle Theorem is true, then all triangles have zero defect. [Hint: For an arbitrary triangle, ABC, draw a line through C parallel to side AB. Justify why you can do this.] 5) Consider the following statements: I: If two triangles are congruent, then they have equal defect. II: If two triangles are similar, then they have...