Show that if the Pythagorean equation holds for all right tri- angles and if XC is...
Fantastic news! We've Found the answer you've been seeking!
Question:
Transcribed Image Text:
Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB = 2IJ can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) D E Figure 6.15 Saccheri quadrilateral associated to AABC. Saccheri triangle plane, that AD= CF = BE, hence that EDAB is a s quadrilateral with base ED, summit AB. Show that a r and its associated Saccheri quadrilateral have equal content- polygonal pieces and then reassemble these pieces to construct i.e., that you can dissect the Saccheri quadrilateral region into the triangular region. ← L Prove that the perpendicular bisector of AB is also perpendi Hence if the plane is hyperbolic, II is divergently parallel to cular to J. (Hint: Use a result about Saccheri quadrilaterals.) AB. Assume now the plane is real, so lengths can be assigned Prove that ED 21J. Deduce that AB > 21J (respectively AB = (Theorem 4.3) and the Saccheri-Legendre theorem applies. 2U) if the plane is hyperbolic (respectively is Euclidean). (d) Prove that K, F, and C are collinear if and only if AC BC (isosceles triangle). If that is the case, prove that F is the mid- point of CK iff the plane is Euclidean. If K, F, and C are not collinear and the plane is not Euclidean, prove that CF is not perpendicular to AB (ray CF does intersect AB at some point G in the case shown, where the angles at A and B are acute, by the crossbar theorem, but CG is not an altitude of the tri- angle if the plane is not Euclidean). 20 Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB 21J can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) = The remaining exercises are in hyperbolic geometry. You can use the results proved in this chapter as well as any results proved in neutral geometry in previous chapters. Do not use any of the Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB = 2IJ can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) D E Figure 6.15 Saccheri quadrilateral associated to AABC. Saccheri triangle plane, that AD= CF = BE, hence that EDAB is a s quadrilateral with base ED, summit AB. Show that a r and its associated Saccheri quadrilateral have equal content- polygonal pieces and then reassemble these pieces to construct i.e., that you can dissect the Saccheri quadrilateral region into the triangular region. ← L Prove that the perpendicular bisector of AB is also perpendi Hence if the plane is hyperbolic, II is divergently parallel to cular to J. (Hint: Use a result about Saccheri quadrilaterals.) AB. Assume now the plane is real, so lengths can be assigned Prove that ED 21J. Deduce that AB > 21J (respectively AB = (Theorem 4.3) and the Saccheri-Legendre theorem applies. 2U) if the plane is hyperbolic (respectively is Euclidean). (d) Prove that K, F, and C are collinear if and only if AC BC (isosceles triangle). If that is the case, prove that F is the mid- point of CK iff the plane is Euclidean. If K, F, and C are not collinear and the plane is not Euclidean, prove that CF is not perpendicular to AB (ray CF does intersect AB at some point G in the case shown, where the angles at A and B are acute, by the crossbar theorem, but CG is not an altitude of the tri- angle if the plane is not Euclidean). 20 Show that if the Pythagorean equation holds for all right tri- angles and if XC is a right angle, then AB 21J can be proved. Deduce from part (c) that such a plane must be Euclidean. (Use these results to add to your answers to Exercise 1.) = The remaining exercises are in hyperbolic geometry. You can use the results proved in this chapter as well as any results proved in neutral geometry in previous chapters. Do not use any of the
Expert Answer:
Related Book For
Posted Date:
Students also viewed these accounting questions
-
With reference to Exercise 8.2, show that if the two samples come from normal populations, then 1 2 is a random variable having a normal distribution with the mean 1 2 and the variance 21/n1 +...
-
Show that if the limiting conditions n , 0, while n. remains constant, are applied to the momentgenerating function of the binomial distribution, we get the moment-generating function of the...
-
Show that if the linear equations x1 + kx2 = c and x1 + x2 = d have the same solution set then the equations are identical.
-
Data Set 32 "Airport Data Speeds" in Appendix B includes Sprint data speeds (mbps). The accompanying TI-83 / 84 Plus display results from using those data to test the claim that they are from a...
-
Suppose that we have a multi-programmed computer in which each job has identical characteristics. In one computation period, T, for a job, half the time is spent in I/O and the other half in...
-
Kindred Nursing Centers East, LLC, owns and operates Whitesburg Gardens, a long-term care and rehabilitation facility, in Huntsville, Alabama. Lorene Jones was admitted to the facility following...
-
For each of the following sets of numbers, calculate the sample size \((N)\), the mean \(\mathrm{X}^{-}\) , the standard deviation \((s)\), and the standard error of the mean \(s \mathrm{X}^{-}\): a....
-
LMC, Inc., is equally owned by Larry, Maurice, and Charles. The owners are sports agents. LMC's income consists solely of fees from the owners' clients. During the current year, LMC's net income from...
-
Suppose that a fund manager has a portfolio of stocks currently worth (in total) 100,000 and that on the market there are a call and a put option contracts available on the same assets of the...
-
Following are separate financial statements of Michael Company and Aaron Company as of December 31, 2013 (credit balances indicated by parentheses). Michael acquired all of Aarons outstanding voting...
-
Rosedale clothiers have decided to make significant reinvestments in its operations. As a result, it will be suspending dividend payments for 2 years. It anticipates paying a dividend again in year 3...
-
Cast Iron Grills, Incorporated, manufactures premium gas barbecue grills. The company reports inventory and cost of goods sold based on calculations from a LIFO periodic inventory system. Cast Iron's...
-
Clopack Company manufactures one product that goes through one processing department called Mixing. All raw materials are introduced at the start of work in the Mixing Department. The company uses...
-
In 2024, the Westgate Construction Company entered into a contract to construct a road for Santa Clara County for $10,000,000. The road was completed in 2026. Information related to the contract is...
-
O'Brien Company manufactures and sells one product. The following information pertains to each of the company's first three years of operations: Variable costs per unit: Manufacturing: Direct...
-
Marshall Company is a large manufacturer of office furniture. The company has recently adopted lean accounting and has identified two value streams-office chairs and office tables. Total sales in the...
-
8. Write a python program If CGPA greater than or equal to 3.75 then Outstanding If CGPA lesser than 3.75 but greater than or equal to 3.5 Excellent If CGPA lesser than 3.5 but greater than or equal...
-
Q:1 Take any product or service offered in Pakistan and apply all determinents of customer Perceived value ?
-
What would the aggregate supply curve look like if a rise in aggregate demand led to a rise in Real GDP but to no change in the price level?
-
1. Explain what the Lorenz curve shows. 2. Describe what the Lorenz curve would show if there were perfect income equality. 3. Explain what the Gini coefficient measures. 4. Describe the relationship...
-
What is the difference between supply and quantity supplied?
-
The position of a particle undergoing simple harmonic motion is given by \(x(t)=20 \cos (8 \pi t)\), where \(x\) is in millimeters and \(t\) is in seconds. For this motion, what are the (a)...
-
Fill in the blanks to make the following statements correct. a. It is difficult to compare two or more data series when absolute numbers and different units are used. For that reason we construct...
-
Fill in the blanks to make the following statements correct. a. The term quantity demanded refers to ___________ purchases by consumers, whereas quantity exchanged refers to ___________ purchases by...
Study smarter with the SolutionInn App