Given that the perpendicular bisector of DC and AB meets at P, and DAB is an obtuse angle. How to prove that DP will never meet AB?

by Level 1 User (280 points)
reopened by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

Best answer

One way to solve this (not necessarily the best way) is to use graphical geometry. In a Cartesian frame of reference let C be (0,0), CB lies along the y-axis, and CD along the x-axis, then let:

A=(a₁,a₂), B=(0,b₂), D=(d₁,0), so C'=(½d₁,0), where b₂>a₂ and d₁>a₁.

CD is y=0 (x-axis), perpendicular bisector of CD is x=d₁/2.

Slope of AB=(a₂-b₂)/a₁, slope of perpendicular is a₁/(b₂-a₂). Because b₂>a₂, the slope of AB is negative, and the slope of the perpendicular is positive.

AB is y=(a₂-b₂)x/a₁+b₂,

perpendicular bisector of AB is y=a₁(x-a₁/2)/(b₂-a₂)+(a₂+b₂)/2.

P is the intersection of the two perpendicular bisectors:

(d₁/2,a₁(d₁/2-a₁/2)/(b₂-a₂)+(a₂+b₂)/2).

Slope of DP is 

s₁=(a₁(d₁/2-a₁/2)/(b₂-a₂)+(a₂+b₂)/2)/(-d₁/2), so s₁<0.

Slope of AD is s₂=a₂/(a₁-d₁), so s₂<0 because d₁>a₁.

Both slopes are negative. If |s₁|>|s₂| then A is to the left of DP, implying that DP does not meet (intersect) AB.

Now, let’s look at the Cosine Rule and apply it to triangle ABD, by joining B and D: BD²=AB²+AD²-2AB.ADcosDAB.

If DAB is obtuse, cosDAB<0, implying BD²>AB²+AD².

BD²=BC²+CD²=b₂²+d₁²; AB²=a₁²+(b₂-a₂)²; AD²=(d₁-a₁)²+a₂².

Therefore, b₂²+d₁²>a₁²+(b₂-a₂)²+(d₁-a₁)²+a₂².

Expanding:

b₂²+d₁²>a₁²+b₂²-2a₂b₂+a₂²+d₁²-2a₁d₁+a₁²+a₂²,

0>2a₁²-2a₂b₂-2a₁d₁+2a₂²,

0>a₁²-a₂b₂-a₁d₁+a₂²,

0>-a₁(d₁-a₁)-a₂(b₂-a₂),

a₁(d₁-a₁)+a₂(b₂-a₂)>0, which we know to be true from given initial conditions (definition of the vertices A, B, C, D). In other words, when d₁>a₁ and b₂>a₂, angle DAB is always obtuse. For P to lie above AB, that is, above the quadrilateral ABCD (as shown in the given figure), where C is the origin of the frame of reference and BC is along the positive y-axis, while AD is along the x-axis, it follows that b₂>a₂.

The slope of BD is -b₂/d₁ and the slope of AB is -(b₂-a₂)/a₁. If A is to be located above diagonal BD so that ABCD is a convex quadrilateral, as shown in the given figure, then (b₂-a₂)/a₁<b₂/d₁.

If DAB is acute, cosDAB>0, implying BD²<AB²+AD², so a₁(d₁-a₁)+a₂(b₂-a₂)<0, that is, a₂(b₂-a₂)<a₁(a₁-d₁). This time, a₁-d₁ is positive implying A is to the right of D, meaning that DP intersects AB. This can be written (b₂-a₂)/a₁<a₁-d₁.

The implication is that for DAB to be obtuse, A must be located to the left of D (a₁<d₁) so that DP cannot meet AB.

 

by Top Rated User (1.1m points)
selected by

Related questions

1 answer
asked Sep 20, 2014 in Word Problem Answers by jugnu Level 1 User (320 points) | 400 views
1 answer
asked Jun 23, 2019 in Word Problem Answers by anonymous | 295 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method proving trigonometric identities least common multiple factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: mathhomeworkanswers.org ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than geometry division fractions how to find x intercept differentiation exponents 8th grade math simplifying fractions geometry 10th grade equivalent fractions inverse function area of a triangle elimination method story problems standard deviation integral ratios simplify systems of equations containing three variables width of a rectangle percentages area of a circle circumference of a circle place value solving triangles parallel lines mathematical proofs solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems quadratic functions number of sides of a polygon length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets set builder notation please help me to answer this step by step significant figures simplifying radicals arithmetic sequences median age problem trigonometry graphing derivatives number patterns adding fractions radicals midpoint of a line roots of polynomials product of two consecutive numbers limits decimals compound interest please help pre-algebra problems divisibility rules graphing functions subtracting fractions angles numbers discrete mathematics volume of a cylinder simultaneous equations integration probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 matrices logarithms how to complete the square mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,441 questions
99,039 answers
2,422 comments
16,939 users