# 6th Grade - Relationships In Geometry

 Grade Level: 6th Skill: Geometry Topic: Relationships in Geometry Goal: Express in symbolic form simple relationships arising from geometry. Skill Description: Use variables (symbols/letters), along with formulas to write expressions or equations to show relationships in geometric shapes (polygons and circles). Understand that a set formula will always be used to solve problems involving: perimeter, area or circumference. Memorize and use the formulas for perimeter; area of a circle, square, rectangle, and triangle; and circumference of a circle. Use these formulas and the relationships between the parts of the one geometric figure or congruent and similar figures to write expressions or equations to show relationships. Precisely describe, classify, and understand relationships among types of geometric figures using their defining properties. Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar and congruent objects. Create and analyze expressions and equations concerning geometric ideas and relationships, such as congruence, similarity. Understand and use academic vocabulary for relating geometric figures: congruent – two or more figures that are the same size, shape, and angle measures similar – two or more figures that have the same shape and angle measures, but different sizes corresponding sides – two proportional sides of congruent or similar figures with relative positions on the figures. corresponding angles – two proportional angles of congruent or similar figures with relative positions on the figures.

## Building Blocks/Prerequisites

### Sample Problems

 (1) A rectangle can be cut into two congruent triangles by drawing a straight line diagonally from a top corner to a bottom corner. Knowing this, if a triangle has the area of n, write an algebraic expression to show how to find the area of one of the triangles. (n/2) (2) Triangle 1 is similar to triangle 2. If triangle 1 has a perimeter measuring x, and triangle 2 is 3 times larger, what expression could be used to find the perimeter of triangle 2? (a) P = 2 x 3 (b) P = 2x (c) P = 3x (c) (3) A rectangle has the width measuring, w. The length of the rectangle is 2 times the width. If the width measures w, write an expression to find the length. (2w) (4) Two congruent squares have sides labeled x and y. The area of square x can be found by using the equation x2. Write an equation to show the area for square y in relationship to square y. (y2) (5) A trapezoid has base 1 with a measurement of n. Base 2 is twice as large as base 1. The height of the figure is 4. Use this information to write an expression to solve for the area of the trapezoid. ((n+2n) x 4/2 or 1/2(n+2n) x 4)

### Online Resources

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### Extra Help Problems

 (1) There are two similar rectangles. The first with a width of W and a length of L. The second triangle is 3 times larger than the first. Write an expression to show how to find the perimeter of the first triangle. (2W + 2L) (2) There are two similar rectangles. The first with a width of W and a length of L. The second triangle is 4 times larger than the first. Write an expression to show how to find the perimeter of the second triangle. 3 (2W + 2L) (3) There are two similar rectangles. The first with a width of W and a length of L. The second triangle is ½ the size of the first. Write an expression to show how to find the area of the first triangle. (A=L x W) (4) There are two similar rectangles. The first with a width of W and a length of L. The second triangle is ½ the size of the first. Write an expression to show how to find the area of the second triangle. (A= ½ L x W) (5) A circle has the diameter, d. Write a formula to show how to find the area of this circle. (C  d) (6) A circle has the radius, r. Express algebraically how to find the circumference of this circle. (C   x 2 x r) (7) There is a trapezoid with the following measures: base 1 - s; base 2 – is 3 times larger than base 1, height – h. Which formula could be used to find area? (a) ½ (s + 3s) x h (b) ½ (s + 3)h (c) s x 3 x h (a) (8) Write an expression to find the perimeter for a trapezoid where b1 = a, b2 = b, h= ½ the size of a. ((a + b)x 1/2a/2 or ½(a + b) x 1/2a) (9) There are two congruent squares. Square One has a side = x and Square two has a size = y. Write an equation to compare the areas of both squares. (x2 = y2 ) (10) A rectangle has a width (w) ¼ the size of its length (L). Which formula could be use to show perimeter? (a) P = ¼ w + L (b) P = (1/4 L 2) + L2 (c) (2L)  (2w 4) (c) (11) The base of a triangle is 5 times its height. Write an expression to show how to find the area. (1/2 (5h x h) (12) A rectangle has a length that is twice as great as its width. Write an expression to solve for area. (W x 2W) (13) The sides of a triangle are w = side one; side two is 10 less than w; and side three is 11 greater than w, write an equation to show the value of side two, if side two is x. (x = w – 10) (14) A rectangle has a perimeter of 100. Write a formula to show perimeter, if length is 9 units more than the width. (2w + (9w x 2) = 100) (15) The sides of a triangle are w = side one; side two is 10 less than w; and side three is 11 greater than w, write an equation to show the value of side three, if side three is y. (y = w + 11) (16) A scalene triangle has a side, b. A similar triangle has a corresponding side, e. Side e is 1/3 the side of side b. Write an expression find the measure of side b. (3e = b) (17) A scalene triangle has a side, b. A similar triangle has a corresponding side, e. Side e is 1/3 the side of side b. Write an expression find the measure of side e. (1/3b = e) (18) A trapezoid has an angle, m. A similar trapezoid has a corresponding angle, p. Angle m is 4 times the size of angle p. Write an expression to show the measure of angle m. (1/4p) (19) A trapezoid has an angle, m. A similar trapezoid has a corresponding angle, p. Angle m is 4 times the size of angle p. Write an expression to show the measure of angle p. (4m) (20) A rectangle has the perimeter of n. Dave is going to cut the rectangle into two equal triangles. Write an expression to show how to find the area of the triangles. ((a x b)/2 for one triangle, a x b for both.)