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Need help please. Just need my answers checked. Thanks in advance. My answers are in brackets.

1. The coordinates of the vertices of ΔJKL are J(-5, -1) , K(0, 1) , and L(2, -5). Which statement correctly describes whether ΔJKL is a right triangle?
a. ΔJKL is a right triangle because JK is perpendicular to JL.
b. ΔJKL is a right triangle because JL is perpendicular to KL.
c. ΔJKL is a right triangle because JK is perpendicular to KL.
[ d. ΔJKL is not a right triangle because no two of its sides are perpendicular. ]

2. The coordinates of the vertices of ΔJKL are J(0, 2) , K(3, 1) , and L(1, -5). Drag and drop the choices into each box to correctly complete the sentences.
The slope of JK is [ -¹/₃ ], the slope of KL is [ 3 ], and the slope of JL is [ -7 ]. ΔJKL [ is ] a right triangle because [ two of these slopes have a product of -1 ].
answer choices: -3 ; 3 ; -7 ; -¹/₃ ; ¹/₇ ; is ; is not ; two of these slopes have a product of -1 ; no two of these slopes have a product of -1

3. The coordinates of the vertices of quadrilateral DEFG are D(-2, 5) , E(2, 4) , F(0, 0) , and G(-4, 1). Which statement correctly describes whether quadrilateral DEFG is a rhombus?
a. Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length.
[ b. Quadrilateral DEFG is not a rhombus because there is only one pair of opposite sides that are parallel. ]
c. Quadrilateral DEFG is not a rhombus because opposites sides are parallel but the four sides do not all have the same length.
d. Quadrilateral DEFG is not a rhombus because there are no pairs of parallel sides.

The slopes of the sides are:
DE = -¹/₄
EF = 2
FG = -¹/₂
GD = 2
I'm debating between A and B.

4. The coordinates of the vertices of quadrilateral ABCD are A(-4, -1) , B(-1, 2) , C(5, 1) , and D(1, -3). Drag and drop the choices into each box to correctly complete the sentences.

The slope of AB is [ 1 ], the slope of BC is [ -¹/₆ ], the slope of CD is [ 1 ], and the slope of AD is [ -²/₅ ]. Quadrilateral ABCD [ is not ] a parallelogram because [ only one pair of opposite sides is parallel ].
answer choices: -²/₅ ; -¹/₆ ; 1 ; ³/₂ ; is ; is not ; both pairs of opposite sides are parallel ; only one pair of opposite side is parallel ; neither pair of opposite sides is parallel

5. The coordinates of the vertices of quadrilateral PQRS are P(-4, 2) , Q(3, 4) , R(5, 0) , and S(-3, -2). Which statement correctly describes whether quadrilateral PQRS is a rectangle?
a. Quadrilateral PQRS is not a rectangle because it has only two right angles.
b. Quadrilateral PQRS is a rectangle because it has four right angles.
c. Quadrilateral PQRS is not a rectangle because it has only one right angle.
d. Quadrilateral PQRS is not a rectangle because it has no right angles.
I'm not sure about this one.

Thanks again.

Respuesta :

Answer:

#1) d. ΔJKL is not a right triangle because no two of its sides are perpendicular; #2) -1/3, 3, -7, is, two of these slopes have a product of -1; #3) a. Quadrilateral DEFG is a rhombus because opposite sides are parallel and all four sides have the same length; #4) 1, -1/6, 1, -2/5, is not, only one pair of opposite sides is parallel; #5) c. Quadrilateral PQRS is not a rectangle because it has only one right angle.

Step-by-step explanation:

#1) The slope of any line segment is found using the formula

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

For JK, this gives us (1-1)/(-5-0) = 0/-5 = 0.  For KL this gives us (1--5)/(0-2) = 6/-2 = -3.  For LJ this gives us (-5-1)/(2--5) = -6/7.  None of these slopes are negative reciprocals, so none of the angles are right angles and this is not a right triangle.

#2) The slope of JK is (2-1)/(0-3) = 1/-3 = -1/3.  The slope of KL is (1--5)/(3-1) = 6/2 = 3.  The slope of LJ is (2--5)/(0-1) = 7/-1 = -7.  Two of these slopes have a product of -1, 3 and -1/3.  This means they are negative reciprocals so this has a right angle; this means JKL is a right triangle.

#3) The slope of DE is (5-4)/(-2-2) = 1/-4 = -1/4.  The slope of EF is (4-0)/(2-0) = 4/2 = 2.  The slope of FG is (0-1)/(0--4) = -1/4.  The slope  of GD is (1-5)/(-4--2) = -4/-2 = 2.  Opposite sides have the same slope so they are parallel.

Next we use the distance formula to find the length of each side:

[tex]d=\sqrt{(y_2-y_1)^2+(x_2-x_1)^2}[/tex]

Using our points, the length of DE is

[tex]\sqrt{(5-4)^2+(-2-2)^2}=\sqrt{1^2+(-4)^2}=\sqrt{1+16}=\sqrt{17}[/tex]

The length of EF is

[tex]d=\sqrt{(4-0)^2+(2-0)^2}=\sqrt{4^2+2^2}=\sqrt{16+4}=\sqrt{20}[/tex]

The length of FG is

[tex]d=\sqrt{(0-1)^2+(0--4)^2}=\sqrt{(-1)^2+(4)^2}=\sqrt{1+16}=\sqrt{17}[/tex]

The length of GD is

[tex]d=\sqrt{(1-5)^2+(-4--2)^2}=\sqrt{(-4)^2+(-2)^2}=\sqrt{16+4}=\sqrt{20}[/tex]

Opposite sides have the same length and are parallel, so this is a parallelogram.

#4) The slope of AB is (-1-2)/(-4--1) = -3/-3 = 1.  The slope of BC is (2-1)/(-1-5) = 1/-6 = -1/6.  The slope of CD is (1--3)/(5-1) = 4/4 = 1.  The slope of DA is (-3--1)/(1--4) = -2/5.  Only one pair of opposite sides is parallel, so this is not a parallelogram.

#5) The slope of PQ is (2-4)/(-4-3) = -2/-7 = 2/7.  The slope of QR is (4-0)/(3-5) = 4/-2 = -2.  The slope of RS is (0--2)/(5--3) = 2/8 = 1/4.  The slope of SP is (-2-2)/(-3--4) = -4/1 = -4.  Only one pair of sides has slopes that are negative reciprocals; this means this figure only has 1 right angle, so it is not a rectangle.

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