Using Algebra in Geometry Practice Questions

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Using Algebra in Geometry Practice Questions

Practice

Answer each of the following questions.

1. The perimeter of a square is (10x + 2) feet. If one side of the square is (x + 5) feet, what is the value of the perimeter of the square?
2. The perimeter of a rectangle is 26 centimeters. If the length of the rectangle is 7 less than 4 times the width of the rectangle, what is the width of the rectangle?
3. Find the area of a rectangle that has a length of (2x + 1) units and a width of (3x – 4) units in terms of x.
4. The base and height of a triangle are two consecutive even numbers. If the area of the triangle is 40 cm², and the base is greater than the height, what is the base of the triangle?
5. If a circle has an area of (x² + 10x + 25)π square feet, what is the radius of the circle?
6. A cylinder has a volume of 729π millimeters³. If the radius and the height are the same, what is the measurement of each?
7. The length of a rectangular prism is half its width, which is half its height. If the volume of the prism is 512 cubic meters, what is the height of the prism?

Solutions

1.	The formula for perimeter of a square is P = 4s. The perimeter of the square is (10x + 2) and one side of the square is (x + 5) feet, so substitute (10x+ 2) for P and substitute (x + 5) for s:
	(10x + 2) = 4(x + 5)
	10x + 2 = 4x + 20
	6x + 2 = 20
	6x = 18
	x= 3
	Because x = 3, the perimeter of the square is 10(3) + 2 = 30 + 2 = 32 feet.
2.	The formula for perimeter of a rectangle is P = 2l + 2w. Let x represent the width of the rectangle. The length is 7 less than 4 times the width, which means that it is 4x – 7. Substitute 26 for P, 4x – 7 for l, and x for w:
	26 = 2(4x – 7) + 2(x)
	26 = 8x – 14 + 2x
	26 = 10x – 14
	40 = 10x
	4 = x
	Because x = 4, the width of the rectangle is 4 centimeters.
3.	The formula for area of a rectangle is A = lw. Multiply the length, 2x + 1, by the width, 3x – 4. Use FOIL:
	First: (2x)(3x) = 6x2
	Outside: (2x)(–4) = –8x
	Inside: (1)(3x) = 3x
	Last: (1)(–4) = –4
	6x2 – 8x + 3x – 4 = 6x2 – 5x – 4 square units.
4.	The formula for area of a triangle is A =. The base and the height are two consecutive even numbers. If one of them is x, then the other must be x + 2, because the next even number is exactly 2 greater than x. The base is greater than the height, so substitute x + 2 for b and x for h. Substitute 40 for A:
	40 =(x + 2)(x)
	40 =(x2 + 2x)
	80 = x2 + 2x
	x2 + 2x – 80 = 0
	Factor this quadratic equation and find the solutions for x. The only factors of –80 that have a difference of 2 are –8 and 10:
	(x – 8)(x + 10) = 0
	x = 8, –10
	The height cannot be a negative value, so the height of the triangle must be 8 centimeters. The next consecutive even number is 10, so the base of the triangle is 10 centimeters.
5.	The formula for area of a circle is A = πr2. Substitute (x2 + 10x + 25) π for A:
	(x2 + 10x + 25) π = πr2
	x2 + 10x + 25 = r2
	Factor x2 + 10x + 25. The only factors of 25 that multiply to 25 and add to 10 are 5 and 5:
	(x + 5)(x + 5) = r2
	(x + 5)2 = r2
	x + 5 = r
	The radius of the circle is (x + 5) feet.
6.	The formula for volume of a cylinder is V = πr2h. The radius and the height are the same, so let x represent both of them. Substitute 729π for V:
	729π = π(x2)(x)
	729 = x3
	9 = x
	The radius and the height of the cylinder are each 9 millimeters.
7.	The formula for volume of a rectangular prism is V = lwh. The length of the prism is half its width, which is half its height, so if x represents the length, then 2x is the width and 4x is the height. Substitute 512 for V, x for l, 2x for w, and 4x for h:
	512 = (x)(2x)(4x)
	512 = 8x3
	64 = x3
	4 = x
	The length of the prism is 4 meters, which means that the width is 2(4) = 8 meters and the height is 2(8) = 16 meters.