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6th Edition

Daniel C. Alexander + 1 other

Publisher: Cengage Learning

ISBN: 9781285195698

Chapter 10.4, Problem 19E

Textbook Problem

Use the analytic method to decide what type of triangle is formed when the midpoints of the sides of an isosceles triangle are joined by line segments.

Elementary Geometry for College Students

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Ch. 10.1 - Plot and then label the points A0,-3, B3,-4, C5,...Ch. 10.1 - Give the coordinates of each point A, B, C, D, and...Ch. 10.1 - Find the distance between each pair of points: a...Ch. 10.1 - If the distance between -2, 3 and -2,a is 5 units,...Ch. 10.1 - If the distance between b,3 and 7, 3 is 3.5 units,...Ch. 10.1 - Find an expression for the distance between a,b...Ch. 10.1 - Find the distance between each pair of points: a...Ch. 10.1 - Find the distance between each pair of points: a...Ch. 10.1 - Find the midpoint of the line segment that joins...Ch. 10.1 - Find the midpoint of the line segment that joins...

Ch. 10.1 - Points A and B have symmetry with respect to the...Ch. 10.1 - Points A and B have symmetry with respect to point...Ch. 10.1 - Points A and B have symmetry with respect to point...Ch. 10.1 - Points A and B have symmetry with respect to the...Ch. 10.1 - Points A and B have symmetry with resect to the...Ch. 10.1 - Points A and B have symmetry with respect to the...Ch. 10.1 - Points A and B have symmetry with respect to the...Ch. 10.1 - Points A and B have symmetry with respect to...Ch. 10.1 - Points A and B have symmetry with respect to a...Ch. 10.1 - In Exercises 20 to 22, apply the Midpoint Formula....Ch. 10.1 - In Exercises 20 to 22, apply the Midpoint Formula....Ch. 10.1 - In Exercises 20 to 22, apply the Midpoint Formula....Ch. 10.1 - A rectangle ABCD has three of its vertices at...Ch. 10.1 - A rectangle MNPQ has three of its vertices at M0,...Ch. 10.1 - Use the Distance Formula to determine the type of...Ch. 10.1 - Use the method of Example 4 to find the equation...Ch. 10.1 - Use the method of Example 4 to find the equation...Ch. 10.1 - For coplanar points A,B, and C, suppose that you...Ch. 10.1 - If two vertices of an equilateral triangle are at...Ch. 10.1 - The rectangle whose vertices are A0, 0,Ba,0,Ca,b,...Ch. 10.1 - There are two points on the y-axis that are...Ch. 10.1 - There are two points on the x-axis that are...Ch. 10.1 - The triangle that has vertices at...Ch. 10.1 - Use the boxing method as suggested in Exercise 33,...Ch. 10.1 - Determine the area of ABC if A=2,1,B=5,3,andC is...Ch. 10.1 - Find the area of ABC In Exercise 35, but assume...Ch. 10.1 - Find the exact volume of the solid that results...Ch. 10.1 - Find the exact volume of the solid that results...Ch. 10.1 - Find the exact volume of the solid that results...Ch. 10.1 - Find the exact volume of the solid that results...Ch. 10.1 - Find the exact lateral area of each solid in...Ch. 10.1 - Find the volume of the solid formed when the...Ch. 10.1 - By definition, an ellipse is the locus of points...Ch. 10.1 - By definition, a hyperbola is the locus of points...Ch. 10.1 - Use the Distance Formula to show that the equation...Ch. 10.1 - Use the Distance Formula to show that the equation...Ch. 10.1 - Following a 90 counterclockwise rotation about the...Ch. 10.1 - Consider the point Ca,b. What is the image of C...Ch. 10.1 - Given the point D3, 2, find the image of D after a...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - In Exercises 1 to 8, draw the graph of each...Ch. 10.2 - Which points lie on the graph of 5x-4y=32? a 4,-3...Ch. 10.2 - Which points lie on the graph of 3x-2y=12? a 6, 2...Ch. 10.2 - Find the slopes of the lines containing: a 2,-3...Ch. 10.2 - Find the slopes of the lines containing: a 3,-5...Ch. 10.2 - Find x so that AB- has slope m, where: a A is...Ch. 10.2 - Find y so that CD- has slope m, where: a C is...Ch. 10.2 - Are these points collinear? a A-2,5, B0,2, and...Ch. 10.2 - Are these points collinear? a A-1,-2, B3,2, and...Ch. 10.2 - Parallel lines l1 and l2 have slopes m1 and m2,...Ch. 10.2 - Parallel lines l1 and l2 have slopes m1 and m2,...Ch. 10.2 - Perpendicular lines l1 and l2 have slopes m1 and...Ch. 10.2 - Perpendicular lines l1 and l2 have slopes m1 and...Ch. 10.2 - In Exercises 21 to 24, state whether the lines are...Ch. 10.2 - In Exercises 21 to 24, state whether the lines are...Ch. 10.2 - In Exercises 21 to 24, state whether the lines are...Ch. 10.2 - In Exercises 21 to 24, state whether the lines are...Ch. 10.2 - Find x such that the points Ax,5, B2,3, and C4,-5...Ch. 10.2 - Find a such that the points A1,3, B4,5, and Ca,a...Ch. 10.2 - Find x such that the line through 2,-3 and 3,2 is...Ch. 10.2 - Find x such that the line through 2,-3 and 3,2 is...Ch. 10.2 - In Exercises 29 to 34, draw the line described....Ch. 10.2 - In Exercises 29 to 34, draw the line described....Ch. 10.2 - In Exercises 29 to 34, draw the line described....Ch. 10.2 - In Exercises 29 to 34, draw the line described....Ch. 10.2 - In Exercises 29 to 34, draw the line described....Ch. 10.2 - In Exercises 29 to 34, draw the line described....Ch. 10.2 - Use slopes to decide whether the triangle with...Ch. 10.2 - If A2,2, B7,3, and C4,x are the vertices of a...Ch. 10.2 - If 2,3, 5,-2, and 7,2 are three vertices not...Ch. 10.2 - Three vertices of rectangle ABCD are A-5,1,...Ch. 10.2 - Show that quadrilateral RSTV is an isosceles...Ch. 10.2 - Show that quadrilateral ABCD is a parallelogram.Ch. 10.2 - Quadrilateral EFGH has the vertices E0,0, Fa,0,...Ch. 10.2 - Find an equation involving a, b, c, d, and e if...Ch. 10.2 - Prove that if two nonvertical lines are parallel,...Ch. 10.2 - Prove that if two lines neither horizontal nor...Ch. 10.2 - Where m0 and b0, the graph of y=mx+b along with...Ch. 10.2 - Where m0, a0, and b0, the graph of y=mx+b, and the...Ch. 10.3 - Find an expression for a The distance between a,0...Ch. 10.3 - Find the coordinates of the midpoint of the...Ch. 10.3 - Find the slope of the line containing the points a...Ch. 10.3 - Find the slope of the line that is a parallel to...Ch. 10.3 - In Exercises 5 to 10, the real numbers a, b, c,...Ch. 10.3 - In Exercises 5 to 10, the real numbers a, b, c,...Ch. 10.3 - In Exercises 5 to 10, the real numbers a, b, c,...Ch. 10.3 - In Exercises 5 to 10, the real numbers a, b, c,...Ch. 10.3 - In Exercises 5 to 10, the real numbers a, b, c,...Ch. 10.3 - In Exercises 5 to 10, the real numbers a, b, c,...Ch. 10.3 - In Exercises 11 to 16, supply the missing...Ch. 10.3 - In Exercises 11 to 16, supply the missing...Ch. 10.3 - In Exercises 11 to 16, supply the missing...Ch. 10.3 - In Exercises 11 to 16, supply the missing...Ch. 10.3 - In Exercises 11 to 16, supply the missing...Ch. 10.3 - In Exercises 11 to 16, supply the missing...Ch. 10.3 - In Exercises 17 to 22, draw an ideally placed...Ch. 10.3 - In Exercises 17 to 22, draw an ideally placed...Ch. 10.3 - In Exercises 17 to 22, draw an ideally placed...Ch. 10.3 - In Exercises 17 to 22, draw an ideally placed...Ch. 10.3 - In Exercises 17 to 22, draw an ideally placed...Ch. 10.3 - In Exercises 17 to 22, draw an ideally placed...Ch. 10.3 - In Exercises 23 to 28, find the equation...Ch. 10.3 - In Exercises 23 to 28, find the equation...Ch. 10.3 - In Exercises 23 to 28, find the equation...Ch. 10.3 - In Exercises 23 to 28, find the equation...Ch. 10.3 - In Exercises 23 to 28, find the equation...Ch. 10.3 - In Exercises 23 to 28, find the equation...Ch. 10.3 - The drawing shows isosceles ABC with AC-BC-. a...Ch. 10.3 - The drawing shows parallelogram RSTV. a What type...Ch. 10.3 - Which formula would you use to establish each of...Ch. 10.3 - Which formula would you use to establish each of...Ch. 10.3 - In Exercises 33 to 36, draw and label a...Ch. 10.3 - In Exercises 33 to 36, draw and label a...Ch. 10.3 - In Exercises 33 to 36, draw and label a...Ch. 10.3 - In Exercises 33 to 36, draw and label a...Ch. 10.3 - In RST, SV- bisects RST. Find the coordinates of...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - In Exercises 1 to 17, complete an analytic proof...Ch. 10.4 - Use the analytic method to decide what type of...Ch. 10.4 - Use the analytic method to decide what type of...Ch. 10.4 - Use slopes to verify that the graphs of the...Ch. 10.4 - Use slopes to verify that the graphs of the...Ch. 10.4 - Use the result in Exercise 20 to find the equation...Ch. 10.4 - Use the result in Exercise 20 to find the equation...Ch. 10.4 - Use the Distance Formula to show that the circle...Ch. 10.4 - Use the result in Exercise 24 to find the equation...Ch. 10.4 - Suppose that the circle with center 0, 0 and...Ch. 10.4 - Consider the circle with center h,k and radius...Ch. 10.4 - Would the theorem of Exercise 7 remain true for a...Ch. 10.4 - Complete an analytic proof of the following...Ch. 10.5 - In Exercises 1 to 4, use division to write an...Ch. 10.5 - In Exercises 1 to 4, use division to write an...Ch. 10.5 - In Exercises 1 to 4, use division to write an...Ch. 10.5 - In Exercises 1 to 4, use division to write an...Ch. 10.5 - In Exercises 5 to 8, draw the graph of each...Ch. 10.5 - In Exercises 5 to 8, draw the graph of each...Ch. 10.5 - In Exercises 5 to 8, draw the graph of each...Ch. 10.5 - In Exercises 5 to 8, draw the graph of each...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 9 to 24, find an equation of the line...Ch. 10.5 - In Exercises 25and 26, find the equation of the...Ch. 10.5 - In Exercises 25and 26, find the equation of the...Ch. 10.5 - In Exercises 27 to 32, use graphing to find the...Ch. 10.5 - In Exercises 27 to 32, use graphing to find the...Ch. 10.5 - In Exercises 27 to 32, use graphing to find the...Ch. 10.5 - In Exercises 27 to 32, use graphing to find the...Ch. 10.5 - In Exercises 27 to 32, use graphing to find the...Ch. 10.5 - In Exercises 27 to 32, use graphing to find the...Ch. 10.5 - In Exercises 33 to 38, use algebra to find the...Ch. 10.5 - In Exercises 33 to 38, use algebra to find the...Ch. 10.5 - In Exercises 33 to 38, use algebra to find the...Ch. 10.5 - In Exercises 33 to 38, use algebra to find the...Ch. 10.5 - In Exercises 33 to 38, use algebra to find the...Ch. 10.5 - In Exercises 33 to 38, use algebra to find the...Ch. 10.5 - In Exercises 39 to 42, use substitution to solve...Ch. 10.5 - In Exercises 39 to 42, use substitution to solve...Ch. 10.5 - In Exercises 39 to 42, use substitution to solve...Ch. 10.5 - In Exercises 39 to 42, use substitution to solve...Ch. 10.5 - For ABC, the vertices are A0,0, Ba,0, and Cb,c. In...Ch. 10.5 - For isosceles PNQ, the vertices are P-2a,0, N2a,0,...Ch. 10.5 - In Exercises 45 and 46, complete an analytic proof...Ch. 10.5 - In Exercises 45 and 46, complete an analytic proof...Ch. 10.5 - Describe the steps of the procedure that enables...Ch. 10.5 - Where C is a real number, the lines 3x+2y=C,...Ch. 10.5 - Where a and b are real numbers, a+b,a-2b is the...Ch. 10.6 - In the Cartesian coordinate system below, name the...Ch. 10.6 - In the Cartesian coordinate system below, name the...Ch. 10.6 - Find a direction vector for the line containing...Ch. 10.6 - Find a direction vector for the line containing...Ch. 10.6 - For the line l:x,y,z=2,3,4+n3,-2,5, find a a point...Ch. 10.6 - For the line l:x,y,z=5,3,-2+n1,2,-2, find a a...Ch. 10.6 - Line m is expressed in point form...Ch. 10.6 - In vector form as in Exercises 5 and 6, find an...Ch. 10.6 - In vector form as in Exercises 5 and 6, find an...Ch. 10.6 - In point form as in Exercise 7 find an equation...Ch. 10.6 - In Exercises 11 to 14, find the distance between...Ch. 10.6 - In Exercises 11 to 14, find the distance between...Ch. 10.6 - In Exercises 11 to 14, find the distance between...Ch. 10.6 - In Exercises 11 to 14, find the distance between...Ch. 10.6 - In Exercises 15 to 18, find the midpoint of the...Ch. 10.6 - In Exercises 15 to 18, find the midpoint of the...Ch. 10.6 - In Exercises 15 to 18, find the midpoint of the...Ch. 10.6 - In Exercises 15 to 18, find the midpoint of the...Ch. 10.6 - In Exercises 19 and 20, use the x, y, and...Ch. 10.6 - In Exercises 19 and 20, use the x, y, and...Ch. 10.6 - Which points lie in the plane 2x+3y+4z=24? a 0,0,6...Ch. 10.6 - Which points lie in the plane 2x+y-z=10? a 0,0,10...Ch. 10.6 - In Exercises 23 to 26, find an equation for each...Ch. 10.6 - In Exercises 23 to 26, find an equation for each...Ch. 10.6 - In Exercises 23 to 26, find an equation for each...Ch. 10.6 - In Exercises 23 to 26, find an equation for each...Ch. 10.6 - Which points lie on the line x,y,z=2,-1,5+n1,2,-1?...Ch. 10.6 - Which points lie on the line x,y,z=2,1,-3+n3,2,-4?...Ch. 10.6 - Explain why the lines below are parallel. l1:...Ch. 10.6 - Explain why the lines below intersect. l1:...Ch. 10.6 - Explain why the lines below are coincident. l1:...Ch. 10.6 - Explain why the lines below are skew. l1:...Ch. 10.6 - For Exercises 33 and 34, apply the following...Ch. 10.6 - For Exercises 33 and 34, apply the following...Ch. 10.6 - The planes x+2y+3z=12 and 2x+3y+z=18 intersect in...Ch. 10.6 - The planes x+2y+z=9 and 3x+y-2z=2 intersect in a...Ch. 10.6 - The line x,y,z=3,4,5+n3,4,-5 intersects the sphere...Ch. 10.6 - The line x,y,z=0,0,0+n1,1,2 intersects the sphere...Ch. 10.6 - Find the surface area and volume of the sphere...Ch. 10.6 - For the sphers x-12+y+22+z-42=36 and x2+y2+z2=64,...Ch. 10.6 - Does the sphere x2+y2+z2=100 have symmetry with...Ch. 10.6 - Lines l1: x,y,z=2,3,-1+n1,1,2 and l2:...Ch. 10.6 - Lines l1: x,y,z=2,0,3+n2,-3,5 and l2:...Ch. 10.6 - The planes with the equation x+2z=12 and y-3z=6...Ch. 10.6 - The planes with the equation x+3z=12 and y-5z=10...Ch. 10.6 - Determine the point of intersection, if such a...Ch. 10.6 - Determine the point of intersection, if such a...Ch. 10.CR - Find the distance between each pair of points: a...Ch. 10.CR - Find the distance between each pair of points: a...Ch. 10.CR - Find the midpoint of the line segment that joins...Ch. 10.CR - Find the midpoint of the line segment that joins...Ch. 10.CR - Find the slope of the line containing each pair of...Ch. 10.CR - Find the slope of the line containing each pair of...Ch. 10.CR - 2,1 is the midpoint of AB-, in which A has...Ch. 10.CR - The y-axis is the perpendicular bisector of RS-....Ch. 10.CR - If A has coordinates 2,1 and B has coordinates...Ch. 10.CR - If R has coordinates -5,2 and S has coordinates...Ch. 10.CR - Without graphing, determine whether the pair of...Ch. 10.CR - Determine whether the points -6,5, 1,7, and 16,10...Ch. 10.CR - Find x so that -2,3, x,6, and 8,8 are collinear.Ch. 10.CR - Draw the graph of 3x+7y=21, and name the...Ch. 10.CR - Draw the graph of 4x-3y=9 by first changing the...Ch. 10.CR - Draw the garph of y+2=-23x-1.Ch. 10.CR - Write an equation for a the line through 2,3 and...Ch. 10.CR - Show that the triangle whose vertices are A-2,-3,...Ch. 10.CR - Show that the triangle whose vertices are A3,6,...Ch. 10.CR - Show that quadrilateral RSTV with vertices R-5,-3,...Ch. 10.CR - In Exercises 21 and 22, find the intersection of...Ch. 10.CR - In Exercises 21 and 22, find the intersection of...Ch. 10.CR - In Exercises 23 and 24, solve the system of...Ch. 10.CR - In Exercises 23 and 24, solve the system of...Ch. 10.CR - Three of the four vertices of a parallelogram are...Ch. 10.CR - A3,1, B5,9, and C11,3 are the vertices of ABC. a...Ch. 10.CR - In Review Exercise 27 to 30, supply the missing...Ch. 10.CR - In Review Exercise 27 to 30, supply the missing...Ch. 10.CR - In Review Exercise 27 to 30, supply the missing...Ch. 10.CR - In Review Exercise 27 to 30, supply the missing...Ch. 10.CR - A2a,2b, B2c,2d, and C0,2e are the vertices of ABC....Ch. 10.CR - Prove the statements in Review Exercises 32 to 36...Ch. 10.CR - Prove the statements in Review Exercises 32 to 36...Ch. 10.CR - Prove the statements in Review Exercises 32 to 36...Ch. 10.CR - Prove the statements in Review Exercises 32 to 36...Ch. 10.CR - Prove the statements in Review Exercises 32 to 36...Ch. 10.CR - Determine whether ABC, with vertices A0,0,0,...Ch. 10.CR - For the line that contains the points P1-1,2,4 and...Ch. 10.CR - Consider the line that contains the points 2,-3,5...Ch. 10.CR - For the sphere with equation x-12+y+22+z+42=36,...Ch. 10.CT - In the coordinate system provided, give the...Ch. 10.CT - In the coordinate system for Exercise 1, plot and...Ch. 10.CT - Use d=x2-x12+y2-y12 to find the length of CD- as...Ch. 10.CT - In the form x,y, determine the midpoint of CD- as...Ch. 10.CT - Complete the following table of x and...Ch. 10.CT - Using the table from Exercise 5, sketch the graph...Ch. 10.CT - Find the slope m of a line containing these...Ch. 10.CT - Line l has slope m=23. Find the slope of any line...Ch. 10.CT - What type of quadrilateral ABCD is represented if...Ch. 10.CT - For the quadrilateral ABCD of Exercise 9 to be a...Ch. 10.CT - Being as specific as possible, describe the type...Ch. 10.CT - Which formula by name is used to establish that a...Ch. 10.CT - Using as few variables as possible, state the...Ch. 10.CT - For providing the theorem The midpoint of the...Ch. 10.CT - In the figure, we see that mRS-=mVT-=0. Find the...Ch. 10.CT - In the form, y=mx+b, find the equation of the line...Ch. 10.CT - Use y-y1=mx-x1 to find the equation of the line...Ch. 10.CT - Use the graphs provided to solve the system...Ch. 10.CT - Use algebra to solve the system consisting of the...Ch. 10.CT - Use the drawing provided for the proof of the...Ch. 10.CT - In cartesian space, line segment P1P2- has...Ch. 10.CT - In cartesian space, the point form of a line is...

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In the following exercises, evaluate the integral using area formulas. 81. 154 ( x3 )2dx

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In Problems 41â€“46 use Theorem B.3 to find Aâ€“1 for the given matrix or show that no inverse exists.
46.

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Use the following data to answer the next five exercises: A pair of studies was performed to measure the effect...

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Systems That May Be Solved by Hand or With Technology The system of equations in Exercises S-7 through S-22may ...

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