19.Marvin is working on a problem for his geometry class. He has to determine if quadrilateral ABCD is a parallelogram, and if so, what kind.

He determines that the diagonals of the quadrilateral have the following characteristics:

- the slopes are opposite reciprocals

- the diagonals have the same midpoint

- the diagonals are congruent

- the diagonals are perpendicular

How would you classify the shape?

square

Rhombus

Rectangle

Kite

Step-by-step explanation:

The parallelogram that has those characteristics is the square.

The square is perfectly symmetric, both diagonals are congruent, because all sides of the square are congruent. Also, the diagonals bisects each angle of the square, creating triangles. Those triangles have 45° angles at their base and the third-angle is 90°, that is, it's a right triangle, where that right angle is between diagonals.

Therefore, a square has congruent and perpendicular diagonals, they have the same midpoint, and their slopes are opposite reciprocals of each other because that's the definition of perpendicularity.

Therefore, the right answer is square.

when functions are first introduced, you will probably have some simplistic "functions" and relations to deal with, usually being just sets of points. these won't be terribly useful or interesting functions and relations, but your text wants you to get the idea of what the domain and range of a function are. small sets of points are generally the simplest sorts of relations, so your book starts with those.

step-by-step explanation:

state the domain and range of the following relation. is the relation a function?

{(2, –3), (4, 6), (3, –1), (6, 6), (2, 3)}

the above list of points, being a relationship between certain x's and certain y's, is a relation. the domain is all the x-values, and the range is all the y-values. to give the domain and the range, i just list the values without duplication:

domain: {2, 3, 4, 6}

range: {–3, –1, 3, 6}

(it is customary to list these values in numerical order, but it is not required. sets are called "unordered lists", so you can list the numbers in any order you feel like. just don't duplicate: technically, repetitions are okay in sets, but most instructors would count off for this.)

while the given set does indeed represent a relation (because x's and y's are being related to each other), the set they gave me contains two points with the same x-value: (2, –3) and (2, 3). since x = 2 gives me two possible destinations (that is, two possible y-values), then this relation is not a function.

note that all i had to do to check whether the relation was a function was to look for duplicate x-values. if you find any duplicate x-values, then the different y-values mean that you do not have a function. remember: for a relation to be a function, each x-value has to go to one, and only one, y-value.

state the domain and range of the following relation. is the relation a function?

{(–3, 5), (–2, 5), (–1, 5), (0, 5), (1, 5), (2, 5)}

i'll just list the x-values for the domain and the y-values for the range: