Our friend claims that a rectangle with the same perimeter as △QRS will also have the same area. Is your friend correct? Explain
. yes; Any rectangle with a perimeter of 12 units will have an area of 6 units. yes; If a rectangle does not have the same perimeter as △QRS, it will not have the same area, so your friend's statement is correct by counterexample. no; A rectangle cannot have a perimeter of 12 units and an area of 6 square units. no; A 1×5 rectangle has a perimeter of 12 units and does not have an area of 6 units, so a counterexample shows that your friend's statement is not correct.
1 answer:
Answer:
A rectangle cannot have a perimeter of 12 units and an area of 6 square units. no
Step-by-step explanation:
We are given vertices in the attached
To solve this question, we are given vertices, we solve using the formula:
√(x2 - x1)² +(y2 - y1)² when given (x1, y1) and (x2, y2)
We are given Triangle QRS
Where: Q = (-2, 1) , R = (-2, -2), S =(2, -2)
QR : Q = (-2, 1) , R = (-2, -2)
= √(-2 - (-2))² + (-2 - 1)²
= √0² + 3²
= √9
= 3 units
RS : R = (-2, -2), S =(2, -2)
= √(2 -(-2))² + (-2 - (- 2))²
= √4² + (0)2
= √16
= 4 units
QS: Q = (-2, 1), S =(2, -2)
= √(2 -(-2))² + (-2 - 1)²
= √(2 + 2)² + (-3)²
= √4² + 9
= √16 + 9
= √25
= 5 units
The Area of Triangle QRS = 1/2bh
b = base = RS = 4 units
h = height = QR = 3 units
= 1/2 × 4 × 3
= 6 square units
Perimeter of Triangle QRS = QR + RS + QS
=( 3 + 4 + 5)units
= 12 units
Our friend is claiming that a rectangle with the same perimeter as will have the same area as the triangle
The perimeter of a rectangle
= 2L + 2W
12 units = 2L + 2W
12 - 2L = 2W
W = 12 - 2L/2
The Area of a Rectangle = L × W
6 square units = LW
6 = LW
6 = L ×(12 - 2L/2)
6 = 12L - 2L²/2
Cross Multiply
6 × 2 = 12L - 2L²
12 = 12L - 2L²
2L² - 12L + 12
2(L² - 6L + 6)
Therefore, from the calculation above, we can conclude:
A rectangle cannot have a perimeter of 12 units and an area of 6 square units. no.
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