Question

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.

Answer 1

The answers provided for ΔJKL, ΔJKL as a right triangle, and quadrilateral ABCD are correct. For quadrilateral DEFG, the answer may be correct depending on side lengths, and for quadrilateral PQRS, slopes must be calculated to determine if it's a rectangle.

Step-by-step explanation:

To determine whether the given shapes are right triangles, rhombuses, rectangles, or parallelograms, we use properties such as the slopes of lines to check for perpendicularity, parallelism, and equal lengths.

For ΔJKL with vertices J(-5, -1), K(0, 1), L(2, -5), you calculated it's not a right triangle because none of the slopes of the sides are negative reciprocals of each other (indicating perpendicular sides). This is correct; no slopes have a product of -1, so answer (d) is correct.

In the second case, ΔJKL with vertices J(0, 2), K(3, 1), L(1, -5) has slopes -1/3, 3, and -7 for sides JK, KL, and JL respectively. Since -1/3 and 3 are negative reciprocals, ΔJKL is a right triangle due to the perpendicular sides JK and KL. So, the filled answers are correct.

Regarding quadrilateral DEFG, you might want to reconsider option (b), as you cannot determine whether it's a rhombus solely based on one pair of opposite sides being parallel. Instead, check if all sides have the same length using the distance formula, and if the opposite sides are parallel by comparing slopes. Since you didn't provide the side lengths, I will only address the slopes here, which you've noted. Answer (b) can be correct if the side lengths are not equal, but if they are, it should be (c).

For quadrilateral ABCD, your assessment of the slopes leads to the conclusion that it is not a parallelogram, which is correct as only one pair of opposite sides (AB and CD having a slope of 1) is parallel. The filled answers are appropriate.

Last, for quadrilateral PQRS, you must calculate the slopes of the sides to determine if the sides are perpendicular to each other by checking if the products of the corresponding slopes are -1. If the slopes of adjacent sides are negative reciprocals of each other, then PQRS has right angles, and it could be a rectangle. Calculate the slopes, and if none are negative reciprocals, then answer (d) is correct stating that PQRS is not a rectangle because it has no right angles.

Answer 2

#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

`m=(y_2-y_1)/(x_2-x_1)`

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:

`d=√((y_2-y_1)^2+(x_2-x_1)^2)`

Using our points, the length of DE is

`√((5-4)^2+(-2-2)^2)=√(1^2+(-4)^2)=√(1+16)=√(17)`

The length of EF is

`d=√((4-0)^2+(2-0)^2)=√(4^2+2^2)=√(16+4)=√(20)`

The length of FG is

`d=√((0-1)^2+(0--4)^2)=√((-1)^2+(4)^2)=√(1+16)=√(17)`

The length of GD is

`d=√((1-5)^2+(-4--2)^2)=√((-4)^2+(-2)^2)=√(16+4)=√(20)`

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.