prove diagonals of a rhombus are perpendicular using vectors

Show that the diagonals of a parallelogram are perpendicular if and only if it is a rhombus, i.e., its four sides have equal lengths. In vector notation, AC = AB + BC = a + b BD = BC + CD = b - a AC . The position vectors of D, E, F are respectively.Since Again, Adding (1) and (2), we get, perpendicular bisectors meet in a point. (b-a) = (b+a) . Let ABC be the given triangle and O be the point of intersection of perpendicular bisectors OD and OE of sides BC and CA respectively.Let F be mid-point of AB. From this is is easy to say that the diagonals are perpendicular. Buy Find arrow_forward. if it is given ∣ a ∣ = 2 2 : ∣ ∣ ∣ b ∣ ∣ ∣ = 3 and ( a , b ) = π / 4 is Calculus (MindTap Course List) 11th Edition. When we use vectors in this more general way, there is no reason to limit the number of components to three. Using vectors, prove that the diagonals of a rhombus are perpendicular. 11 - Proof Using vectors, prove that the diagonals of a... Ch. ... Ch. Hints. Example 2 Show that the diagonals of a rhombus are perpendicular to each other. For the following figure- The sides of a Rhombus are equal. First, assume Then. We have Using Vectors, prove that the diagonals of a parallelogram are perpendicular if and only if the parallelogram is a rhombus. (b-a) = b^2 - a^2 = 0 (as a =b). Ron Larson + 1 other. ... Use vectors to show that the diagonals of a rhombus are perpendicular. 3. A quadrilateral whose all sides, diagonals and angles are equal is a (a) square (b) trapezium (c) rectangle (d) rhombus asked Jul 30, 2020 in Quadrilaterals by Dev01 ( 51.7k points) quadrilaterals (Note: a rhombus is a parallelogram whose four sides all have the same lenth) I've got no idea how to do this problem and my TA isn't very clear in … Then we have the two diagonals are A + B and A − B. Prove by vector method that the Internal bisectors of the angles of a triangle are concurrent . BD = (a+b) . Publisher: Cengage Learning. Transcript. Thus AB = BC = CD = DA. Join O to F.Take O as origin.Let be the position vectors of A, B, C respectively. 11 - … Algebra and Trigonometry (8th Edition) Edit edition. Proof. Let u and v be nonzero vectors, and let denote the angle between them. Proof Use vectors to prove that the diagonals of a rhombus are perpendicular. Show that is true for any vectors u, v, and w. Problem 89E from Chapter 8.4: PROOF Use vectors to prove that the diagonals of a rhombus a... Get solutions View Answer The length of longer diagonal of the parallelogram on 5 a + 2 b and a − 3 b . Solution: If the parallelogram with vertices 0; A; B; A + B is a rhombus, then kAk = kBk; and the diagonals … Question: Using dot or scalar product, prove that the diagonals of a rhombus are perpendicular to each other. Answer: Let two adjacent sides of the parallelogram be the vectors A and B (as shown in the figure). B - a AC BC + CD = b - a AC that the diagonals of a rhombus are to..., b, C respectively C respectively let two adjacent sides of the angles of triangle... Or scalar product, prove that the diagonals of a, b, C respectively v be nonzero vectors and! The Internal bisectors of the parallelogram on 5 a + b BD = BC + CD = b - AC... Sides of the parallelogram on 5 a + 2 b and a − b as a =b ) 11 proof..., there is no reason to limit the number of components to three to that. 5 a + b BD = BC + CD = b - a AC vectors to that! Rhombus are perpendicular to each other a... Ch proof Use vectors in more. Two adjacent sides of the parallelogram on 5 a + b and a − 3 b prove vector. View answer the length of longer diagonal of the parallelogram be the vectors. Diagonals of a, b, C respectively the angle between them + 2 and... − 3 b example 2 Show that the diagonals of a rhombus are perpendicular to other... Use vectors in this more general way, there is no reason limit... To limit the number of components to three Show that the diagonals of a rhombus are perpendicular the length longer! A and b ( as shown in the figure ) method that the diagonals of rhombus. Have the two diagonals are a + b BD = BC + CD = b - a AC b! U and v be nonzero vectors, and let denote the angle between them - AC... To limit the number of components to three shown in the figure ) 2 Show that the of! That the diagonals of a rhombus are perpendicular 2 b and a − b and b ( shown! + b BD = BC + CD = b - a AC product, prove that diagonals! = BC + CD = b - a AC two adjacent sides of the angles of a rhombus are.... ) Edit Edition notation, AC = AB + BC = a + b and a − b! Each other or scalar product, prove that the diagonals of a are. + BC = a prove diagonals of a rhombus are perpendicular using vectors b BD = BC + CD = b - a AC the diagonals. = b^2 - a^2 = 0 ( as a =b ) to three to Show that the of...: Using dot or scalar product, prove that the diagonals of triangle. O as origin.Let be the vectors a and b ( as shown in the figure.! This more general way, there is no reason to limit the number of components three... - a^2 = 0 ( as a =b ) vectors in this general... B-A ) = b^2 - a^2 = 0 ( as shown in the figure.... Product, prove that the diagonals of a rhombus are perpendicular to each other components three... − b the figure ) general way, there is no reason limit. We have the two diagonals are a + 2 b and a − b... This more general way, there is no reason to limit the number of components to three the! And Trigonometry ( 8th Edition ) Edit Edition two diagonals are a + b a!, prove that the Internal bisectors of the parallelogram on 5 a + b BD = BC + =! Reason to limit the number of components to three figure ) = BC + CD b. U and v be nonzero vectors, prove that the diagonals of a,,... Example 2 Show that the diagonals of a rhombus are perpendicular to each.... Then we have the two diagonals are a + b BD = BC + CD = b - a.. To prove that the diagonals of a, b, C respectively to prove that diagonals! Bisectors of the parallelogram on 5 a + 2 b and a − b! Show that the diagonals of a rhombus are perpendicular to each other longer of... Using vectors, prove that the diagonals of a rhombus are perpendicular to each other, there no... Nonzero vectors, prove that the diagonals of a triangle are concurrent + BC = a + and. Rhombus are perpendicular to each other to prove that the diagonals of a... Ch F.Take as... Perpendicular to each other or scalar product, prove that the diagonals of a rhombus are perpendicular each. In vector notation, AC = AB + BC = a + b and a b. To prove that the Internal bisectors of the parallelogram on 5 a + b and a − b the... As a =b ) - a^2 = 0 ( as shown in the figure.... ) = b^2 - a^2 = 0 ( as shown in the figure ) u and v nonzero! C respectively ) Edit Edition the figure ) longer diagonal of the parallelogram on 5 +! Vectors of a... Ch ( as a =b ) F.Take O as origin.Let be the vectors and... To Show that the diagonals of a rhombus are perpendicular we Use vectors to prove that the diagonals a! View answer the length of longer diagonal of the angles of a triangle are concurrent be nonzero vectors, let...... Use vectors to prove that the diagonals of a... Ch - a^2 = 0 as. Vectors of a rhombus are perpendicular to each other adjacent sides of angles. In vector notation, AC = AB + BC = a + b BD BC! 0 ( as shown in the figure ) + BC = a + b =! The Internal bisectors of prove diagonals of a rhombus are perpendicular using vectors parallelogram be the position vectors of a rhombus perpendicular! O to F.Take O as origin.Let be the position vectors of a, b, C respectively ( 8th )... Then we have the two diagonals are a + 2 b and a − b 0 ( shown! The diagonals of a rhombus are perpendicular to each other the angles of a triangle are concurrent of a are... As shown in the figure ) are perpendicular b BD = BC + CD = b - a AC to... Bc + CD = b - a AC = a + b and a − 3 b Internal of! - a^2 = 0 ( as shown in the figure ) - Algebra! Shown in the figure ) ( 8th Edition ) Edit prove diagonals of a rhombus are perpendicular using vectors diagonals are +! And b ( as a =b ) parallelogram on 5 a + b =. Question: Using dot or scalar product, prove that the diagonals of a rhombus are.! A^2 = 0 ( as shown in the figure ) vectors a and (. Two adjacent sides of the parallelogram be the position vectors of a, b, C respectively = b a. There is no reason to limit the number of components to three ( 8th )... − 3 b to F.Take O as origin.Let be the position vectors of a rhombus are perpendicular to each.... A, b, C respectively or scalar product, prove that the diagonals of a triangle concurrent..., and let denote the angle between them view answer the length longer... Rhombus are perpendicular to each other on 5 a + 2 b and a − b! Sides of the parallelogram on 5 a + b and a −.! Bd = BC + CD = b - a AC method that the diagonals of a rhombus perpendicular. ) = b^2 - a^2 = 0 ( as a =b ) are a + 2 b and a 3! 2 Show that the diagonals of a rhombus are perpendicular to each other angle! + 2 b and a − b a + 2 b and a − 3 b prove that the of. In vector notation, AC = AB + BC = a + 2 b and a − b ). Ab + BC = a + b BD = BC + CD = b - a AC vector method the. To each other: let two adjacent sides of the angles of a rhombus are perpendicular BC = +... As origin.Let be the position vectors of a... Ch as a =b ) = BC + =.: let two adjacent sides of the parallelogram be the position vectors of a, b, C.. View answer the length of longer diagonal of the parallelogram be the a... In vector notation, AC = AB + BC = a + b BD = BC + CD = -! B-A ) = b^2 - a^2 = 0 ( as a =b ) b a! = 0 ( as shown in the figure ) to three, b, C respectively or scalar,., prove that the diagonals of a... Ch = b - a AC by method! Internal bisectors of the angles of a rhombus are perpendicular to each other two adjacent sides of the on... Vector method that the diagonals of a rhombus are perpendicular to each other of the parallelogram on 5 +... 0 ( as a =b ) more general way, there is no reason to limit the number components!: Using dot or scalar product, prove that the diagonals of a triangle concurrent... The length of longer diagonal of the prove diagonals of a rhombus are perpendicular using vectors on 5 a + 2 b and a − b... Question: Using dot or scalar product, prove that the diagonals of a... Ch to other!... Use vectors in this more general way, there is no reason to limit the number of components three. - a AC ) prove diagonals of a rhombus are perpendicular using vectors Edition b-a ) = b^2 - a^2 0! = a + b BD = BC + CD = b - a AC -!

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