The ratio of the perimeter of triangle PQR to the perimeter of rectangle ABCD is 5 : 9. All in centimeters
Triangle (sides) PQ: (x-3) PR: (3x-5) QR: (4x+3)
Rectangle (sides) AB: (2x+2) BC: (4x - 1/2) [AB is longer than BC]
a. Write algebraic expressions for the perimeters of triangle PQR and rectangle ABCD.
b. Write a linear equation using the algebraic expressions written in part a. Then solve for x.
c. Find the area of rectangle ABCD.
[ (x-3) + (3x-5) + (4x+3) ] : [2(2x+2)+ 2(4x - 1/2) ] = 4 : 9
(8x-5) / (12x+3) = 4/9
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The ratio of the perimeter of triangle PQR to the perimeter of rectangle ABCD is 5 : 9. All in centimeters
Triangle (sides) PQ: (x-3) PR: (3x-5) QR: (4x+3)
Rectangle (sides) AB: (2x+2) BC: (4x - 1/2) [AB is longer than BC]
a. Write algebraic expressions for the perimeters of triangle PQR and rectangle ABCD.
b. Write a linear equation using the algebraic expressions written in part a. Then solve for x.
c. Find the area of rectangle ABCD.
a. Perimeter of triangle PQR = (x-3) + (3x-5) + (4x+3) = 8x-5 cm
Perimeter of rectangle ABCD = 2(2x+2) + 2(4x - 1/2) = 12x+3 cm
b. The linear equation using the given ratio of perimeters is:
(8x-5)/(12x+3) = 5/9
Cross-multiplying and simplifying, we get:
72x - 45 = 60x + 15
12x = 60
x = 5
Therefore, the value of x is 5.
c. The area of rectangle ABCD is:
Area = AB x BC
Area = (2x+2) x (4x - 1/2)
Area = 8x^2 + 3x - 1 cm^2
Substituting the value of x as 5, we get:
Area = 203 cm^2
Therefore, the area of rectangle ABCD is 203 square centimeters.
a. The perimeter of a triangle is the sum of the lengths of its three sides. So, the expressions for the perimeters of triangle PQR and rectangle ABCD are as follows:
Perimeter of triangle PQR = (x-3) + (3x-5) + (4x+3)
Perimeter of rectangle ABCD = 2(2x+2) + 2(4x - 1/2)
b. To write a linear equation using the algebraic expressions for the perimeters and solve for x, we can use the given ratio. The ratio states that the perimeter of triangle PQR to the perimeter of rectangle ABCD is 5:9. In equation form, we have:
Perimeter of triangle PQR / Perimeter of rectangle ABCD = 5/9
Substituting the expressions from part a, we get:
((x-3) + (3x-5) + (4x+3)) / (2(2x+2) + 2(4x - 1/2)) = 5/9
Simplifying the equation:
(8x-5) / (16x + 3) = 5/9
Cross multiplying:
9(8x-5) = 5(16x + 3)
72x - 45 = 80x + 15
Collecting like terms:
72x - 80x = 15 + 45
-8x = 60
Dividing both sides by -8:
x = -7.5
c. To find the area of rectangle ABCD, we need the lengths of its sides. Substituting x = -7.5 into the expressions for the sides of the rectangle, we get:
Length (AB) = 2x + 2 = 2(-7.5) + 2 = -15 + 2 = -13 cm (assuming negative values are allowed)
Width (BC) = 4x - 1/2 = 4(-7.5) - 1/2 = -30 - 1/2 = -30.5 cm (assuming negative values are allowed)
The area of a rectangle is given by the formula A = length * width. Therefore, the area of rectangle ABCD is:
Area of rectangle ABCD = (-13) * (-30.5) = 396.5 square cm (assuming negative values are allowed)