A formula to solve the question is attached from a book. Thank you. Prove that if a Pythagorean Triangle has even sides, then its perimeter, P, dividesits area, A. last equation can be expressed as(a + B)(a - B) = 1.wwwwAll the numbers concerned are rational, so if the product of two numbers is 1, they areO reciprocals. That is, one number must be m/n and the other n/m, where m and n aret integers. Settinga + B =anda - B =we nd by addition thatmand by subtraction that11 (mB =Consequently,m² + n²m2 -n²wwww(1)2mn2mnBut y = Bx and z = ax; if we now put x = 2mn, so as to get a solution in integers, itfollows that=2mn.y = m-n².z = m? + n2.=%²rwwwwwThese are well-known formulas for nding right triangles with sides of integral length andwere used in Hellenistic times by Diophantus (circa 150), the most original mathematicianof late antiquity.

We have to prove perimeter P divides area A.

Answered: Prove that if a Pythagorean Triangle…

Math

Geometry

Prove that if a Pythagorean Triangle has even sides, then its perimeter, P, divides its area, A.

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter3: Triangles

Section3.4: Basic Constructions Justified

Problem 38E

See similar textbooks

Related questions

Q: A two-column proof can be used to demonstrate that a diagonal splits a parallelogram into two…

Q: Prove if a quadrilateral ABCD is cyclic, then its opposite angles are supplementary.

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: 6 in. 8 in.

Q: Let the line DE be a line parallel to the base BC of triangle ABC. Show that if the line AF divides…

A: Given,DE∥BCAF bisects BCLet intersection of AF with DE be 'x'

Q: 3. Conduct a formal proof to prove that the diagonals of an isosceles trapezoid are congruent.

A: Given: Consider the isosceles trapezoid ABCD shown below: AB=CD To prove: AC≅BD

Q: Prove that the angles at the base of an isosceles triangle are equal.

A: To prove: The angles at the base of an isosceles triangle are equal. Proof: Let an isoscles triangle…

Q: 2. Prove that if the Fermat-Torricelli point of an acute triangle ABC coincides with the incenter,…

Q: If a quadrilateral has EXACTLY 3 congruent sides and the 4th side is parallel to one of the other…

Q: In a right triangle, with legs a and b and hypotenuse c, thePythagorean Theorem states that_________

A: It is given that, In a right triangle with a leg a and b and hypotenuse c.

Q: 1. If the bisectors of angles of a quadrilateral enclose a rectangle, then show that it is a…

A: As per our guideline we are allowed to do 1 question at a time. Please post other questions next…

Q: When the diagonals of rhombus MNPQ are drawn, how do the areas of the four resulting smaller…

A: Characteristics of a rhombus All sides are equal. (i) Diagonals bisect each other at right…

Q: 12 6. 8. A.

A: In ∆ACB and ∆ECD ACBC=CEDC 69=812 23=23∠ACB=∠ECD (Vertical Opposite angle)∴∆ACB ~∆ECD…

Q: side

Q: Prove that if the Fermat-Torricelli with the orthocenter, then ABC is equilateral. point of an acute…

A: To prove that if the fermat-torricelli point of an acute triangle ABC coincides with the…

Q: Prove that any angle bisector of a triangle separates the opposite side into segments whose lengths…

Q: The Pythagorean Theorem states that in any equals triangle, the sum of the squares of the lengths of…

A: The Pythagorean Theorem states that in any ________ triangle, the sum of the squares of the lengths…

Q: if ABCD is a rhombus, AC=4, and ADC is an equilateral triangle, what is the area of rhombus?

A: Drawing the figure as per the question, As we are given that AC=4 and ADC is an equilateral…

Q: if a triangle has sides of x,2x,and 3x-1, what is the perimeter?

A: Here is the solution.

Q: Can you use the ASA Postulate or the AAS Theorem to prove the triangles congruent?

A: When two lines intersect each other than opposite angles are equal.

Q: It is possible to find the perimeter of any given quadrilateral on the coordinate plane. O True O…

Q: Prove that if a line bisects one side of a triangle and is parallel to a second side, then it…

A: Basic proportionality Theorem of triangle is define as if a line drawn parallel to one side of…

Q: James concurs that since D is the midpoint of AB, E is the midpoint of BC, and F is the midpoint of…

A: Here we have, James concurs that since D is the midpoint of AB, E is the midpoint of BC, and F is…

Q: Prove that the measure of an exterior angle of a triangle is equal to the sum of the measures of two…

Q: In a right triangle, with legs a and b and hypotenuse c, thePythagorean Theorem states that .

A: In a right triangle, with legs a and b and hypotenusec, the Pythagorean Theorem states that______

Q: prove: two isosceles triangles are similar if and only if their angles at the vertex are congruent

Q: In Triangle JKL, ZJ is congruent to ZL. K 67.8

A: as ∠J and ∠L are congruent so, ∠J=∠L let's assume ∠J=∠L=θ now at K point…

Q: 4. If AB = AD = y +5, BC = y, and CD =+ 3, find the perimeter of ABCD. %3D

A: 4. The figure with AB = AD = y+5, BC = y and CD = y2+3 is given by To evaluate : The perimeter of…

Q: If the area of a rhombus is 112 cm? and one of its diagonal is 14 cm find its othe

A: topic - area of rhombus from its diagonals

Q: (4) Prove that if two altitudes in a triangle are congruent, then the triangle is Isosceles.

Q: A kite is a quadrilateral with two pairs of adjacent congruent sides. O True O False

Q: Prove that a quadrilateral can be inscribed in a circle if and only if its opposite angles are…

Q: Given Square ABCD and EDC = ZECD, prove AABE is equilateral.

A: So, ABCD is a square, angle EDC and ECD are equal, (let them be x), i.e. E is exactly lies on the…

Q: The diagonal of a parallelogram divides it into two congruent triangles.

Q: Are these triangles similar? If so, which theorem can be used to prove 55° STH-GFH 550 G Submit

Q: A regular octagon can be divided into 8 congruent isosceles triangles, one for each of its sides. O…

A: . A Regular Octagon

Q: Prove the Pythagorean Theorem with Similar Triangles using a two-column proof.

A: To Prove the Pythagorean Theorem with Similar Triangles using a two-column proof

Q: Prove that if two triangles have congruent bases, then their areas are in the same ratio as their…

A: To show that the ratio of the areas is equal to the ratio of the heights, if the two triangles have…

Q: prove that the sum of the interior angles of a triangle ABC is 180 degrees

Q: Can the sides of a triangle have length 0.7,1.1,and 1.8

Q: A square with sides of length 2 in. rests (as shown) on a square with sides of length 6 in. Find the…

A: Given that - AE = EF = FG = AG = 2 inAnd,AH = HC = CD = AD = 6 inWe see that,∠AEB and ∠DEC both are…

Q: If a triangle has sides of length x, x+2, and x-4, what is the perimeter of the triangle in terms of…

A: Given information- We have given the length of sides of a triangle x, x+2, x-4 We have to find the…

Q: Show that if a line is drawn from the vertex of an isosceles triangle that bisects the base of the…

A: Let ABC is an isosceles triangle such that AB=AC. Let AD is the bisector of angle A. To prove BD=BC…

Q: Prove that if a point is contained in two angles of a triangle, then it is contained in the interior…

A: In a triangle, any two angles are adjacent (i.e, sharing a common hand)

Q: Consider any eight points such that no three are collinear.How many triangles are determined?

A: Given: Eight point such that no three are collinear, To determine: The number of triangle that can…

Q: 25. Given: Quadrilateral PQST with midpoints A, B, C, and D of the sides Prove: ABCD is a O

Q: Given an acute-angled triangle ABC, construct a square with one side lying on BC while the other two…

A: Image is attached with detailed explanation.

Q: that if P, Q, R and S form a quadrilateral, then it cannot be cyclic.

A: A quadrilateral inscribed in a circle is known as a cyclic quadrilateral. so, if a pair of…

Q: Given a cyclic quadrilateral whose sides are 4 cm, 5 cm, 8 cm, and 11 cm, its area is?

Q: Draw an inscribed quadrilateral ABCD and its diagonals intersecting at E. Name two pairs of similar…

A: Clearly from the below daigram we can see thatthe two pairs of…

Q: V is a point on segment UW. If Uv=11.3 and VW=8.4, what is UW?

Question

100%

A formula to solve the question is attached from a book. Thank you.

Prove that if a Pythagorean Triangle has even sides, then its perimeter, P, divides
its area, A.

last equation can be expressed as
(a + B)(a - B) = 1.
wwww
All the numbers concerned are rational, so if the product of two numbers is 1, they are
O reciprocals. That is, one number must be m/n and the other n/m, where m and n are
t integers. Setting
a + B =
and
a - B =
we nd by addition that
m
and by subtraction that
1
1 (m
B =
Consequently,
m² + n²
m2 -n²
wwww
(1)
2mn
2mn
But y = Bx and z = ax; if we now put x = 2mn, so as to get a solution in integers, it
follows that
=2mn.
y = m-n².
z = m? + n2.
=%²
r
wwwww
These are well-known formulas for nding right triangles with sides of integral length and
were used in Hellenistic times by Diophantus (circa 150), the most original mathematician
of late antiquity.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

a Is it really necessary to construct all three bisectors of the angles of a triangle to locate its incenter? b Is it really necessary to construct all three perpendicular bisectors of the sides of a triangle to locate its circumcenter?
Given that rectangle ABCE is similar to rectangle MNPR and that CDEPQR, what can you conclude regarding pentagon ABCDE and pentagon MNPQR?
What type of trapezoid a has congruent diagonals. b has congruent base angles.
The bisectors of two consecutive angles of HJKL are shown. What can you conclude regarding P?
a What is the general name of the point of concurrence for the three perpendicular bisectors of the sides of a triangle? b What is the general name of the point of concurrence for the three medians of a triangle?
Two sets of rails railroad tracks are equally spaced intersect but not at right angles. Being as specific as possible, indicate what type of quadrilateral WXYZ is formed.
A trapezoid is a four-sided figure that contains one pair of parallel sides. Which sides of the trapezoid MNPQ appear to be parallel?
Use the results from Exercises 36 and 37 to find the exact length of the radius of the inscribed circle for a triangle with sides of lengths a 8, 15 and 17 b 7, 9 and 12