Answer:
The figure PEST is a rhombus
Step-by-step explanation:
* Lets talk about the difference between all these shapes
- At first to prove the shape is a parallelogram you must have one
of these conditions
# Each two opposite sides are parallel OR
# Each two opposite sides are equal in length OR
# Its two diagonals bisect each other
- After that to prove the parallelogram is:
* A rectangle you must have one of these conditions
# Two adjacent sides are perpendicular to each other OR
# Its two diagonals are equal in length
* A rhombus you must have one of these conditions
# Two adjacent sides are equal in length OR
# Its two diagonals perpendicular to each other OR
# Its diagonals bisect its vertices angles
* A square you must have two of these conditions
# Its diagonals are equal and perpendicular OR
# Two adjacent sides are equal and perpendicular
* Now lets solve the problem
∵ The vertices of the quadrilateral PEST are
P (-1 , -5) , E (8 , 2) , S (11 , 13) , T (2 , 6)
- Lets find the slope from each two points using this rule :
m = (y2 - y1)/(x2 - x1), where m is the slope and (x1 , y1) , (x2 , y2)
are two points on the line
- Let (x1 , y1) is (-1 , -5) and (x2 , y2) is (8 , 2)
∴ m of PE = (2 - -5)/(8 - -1) = 7/9
- Let (x1 , y1) is (8 , 2) and (x2 , y2) is (11 , 13)
∴ m of ES = (13 - 2)/(11 - 8) = 11/3
- Let (x1 , y1) is (11 , 13) and (x2 , y2) is (2 , 6)
∴ m of ST = (6 - 13)/(2 - 11) = -7/-9 = 7/9
- Let (x1 , y1) is (2 , 6) and (x2 , y2) is (-1 , -5)
∴ m of TP = (-5 - 6)/(-1 - 2) = -11/-3 = 11/3
∵ m PE = m ST = 7/9
∴ PE // ST ⇒ opposite sides
∵ m ES = m TP = 11/3
∴ ES // TP ⇒ opposite sides
- Each two opposite sides are parallel
∴ PEST is a parallelogram
- Lets check if the parallelogram can be rectangle or rhombus or
square by one of the condition above
∵ If two line perpendicular , then the product of their slops = -1
- Lets check the slopes of two adjacent sides (PE an ES)
∵ m PE = 7/9
∵ m ES = 11/3
∵ m PE × m ES = 7/9 × 11/3 = 77/27 ≠ -1
∴ PE and ES are not perpendicular
∴ PEST not a rectangle or a square (the sides of the rectangle and
the square are perpendicular to each other)
- Now lets check the length of two adjacent side by using the rule
of distance between two points (x1 , y1) and (x2 , y2)
d = √[(x2 - x1)² + (y2 - y1)²]
- Let (x1 , y1) is (-1 , -5) and (x2 , y2) is (8 , 2)
∴ PE = √[(8 - -1)² + (2 - -5)²] = √[9² + 7²] = √[81 + 49] = √130 units
- Let (x1 , y1) is (8 , 2) and (x2 , y2) is (11 , 13)
∴ ES = √[(11 - 8)² + (13 - 2)²] = √[3² + 11²] = √[9 + 121] = √130 units
∴ PE = ES ⇒ two adjacent sides in parallelogram
∴ The four sides are equal
* The figure PEST is a rhombus
