Geometry 2

Question 35.A garden gate that is 3 feet wide and 4 feet tall needs a diagonal brace to make it stable. How long apiece of wood will be needed for this diagonal brace?

3^2 + 4^2 = 𝑥^2 9 + 16 = 𝑥^2 25 = 𝑥^2 √25 = 𝑥 𝑥 = 5 A piece of wood of length of 5 ft will be needed for the diagonal brace.

Question 8: Draw a venn diagram or other clear diagram showing the relationships between the sets of squares, rectangles and parallelograms.

All squares are rectangles, and all rectangles ore parallelograms

Question 12: Using a compass and a straightedge, construct an isosceles triangle that is NOT equilateral. Explain how you know that the triangle is isosceles and not equilateral without measuring it

All the points on a circle are the same distance from the center. In this method, the center is one vertex of the triangle. Choosing the other two vertices to be points on the circle ensures that both are the same distance from the first vertex, giving us two sides of the same length

Question 3. Given that the indicated lines in the figure below are parallel, determine angle 𝑎. Explain your reasoning briefly

Angle b is 55° because is opposite to the 55° angle given.Angle c is 125° because is corresponding with the 125° angle given and by the Parallel Postulate.Angle d is 55° because it forms a straight line with angle c.Now, b + d + e = 180° because the sum of the angles in a triangle is 180°, so,55° + 55° + e = 180° → e = 180° - 55° - 55° = 70°.Therefore a = 110° since it forms a straight line with 𝑒

Question 4 Given that the indicated lines in the figure below are parallel, determine angle 𝑎. Explain your reasoning briefly

Angle b is 95° because is corresponding with the 95° angle given and by the Parallel Postulate.Angle c is 85° because it forms a straight line with angle b.Now, c + d + 35° = 180° because the sum of the angles in a triangle is 180°, so, 85° + d + 35° = 180° → d = 180° - 85° - 35° = 60°.Therefore a = 120° since it forms a straight line with d

Question 1. Explain clearly why Keisha's and Aaron's "angle explorers" in the figure below show angles of the same size. Why isn't Keisha's angle bigger? As part of your explanation, discuss how we think about sizes of angles

Angles measurement is defined in terms of rotation abouta fixed point, the vertex. Both angles show the same amount of rotation, the length of the sides of the angle does not affect the amount of rotation about the vertex

Question 33.A concrete patio will be made in the shape of a 12-foot by 12-foot square with half-circles attached at two opposite ends, as shown below. What is the area of this patio?

Area of the patio = A1 + A2 + A3A1 = 12 × 12 = 144 ft2A2 + A3 = Area of a circle with a radius of 6 ftA2 + A3 = 𝜋(6)2 = 36𝜋 ≈ 113 ft2Area of the patio = 144 + 113 = 257 ft2

Question 26.Determine the area of the shaded triangle below. Explain your reasoning

Area rectangle = 6 × 3 = 18 cm2𝐴1 = 12 (4 × 2) = 4 cm2𝐴2 = 12 (6 × 1) = 3 cm2𝐴3 = 12 (2 × 3) = 3 cm2 Shaded area = 18 − 4 − 3 − 3 = 8 cm2

Question 19.Explain why each one of the two blocks in the figure below can be considered the "biggest" of the two by first comparing the blocks' sizes with respect to a 2-dimensional aspect f the blocks, and then comparing the blocks' sizes with respect to a 3-dimensional aspect of the blocks.

Comparing a 2-dimensional aspect of the blocks, like surface area, the first block would be bigger because it has a surface area of 2 ∙ (5 ∙ 5) + 4(1 ∙ 5) = 60 in^2, while the second block has a surface area of 6 ∙ (3 ∙ 3) = 54 in^2.Comparing a 3-dimensional aspect of the blocks, like volume, the second block would be considered bigger because it has a volume of 3 ∙ 3 ∙ 3 = 27 in^3, while the first block has a surface area of 1 ∙ 5 ∙5 = 25 in^3.

Question 30.If you use the diameter of a circle to measure its circumference, what is the result? Does it depend on whether the circle is small or large? Explain

For any circle when the circumference is measured by its diameter the result is always the same number, which is approximately 3.14 (exactly 𝜋). All circles are just scaled versions of each other,either scaled smaller or scaled larger. When we scale shapes, the ratio between the lengths of different parts of the shape remains the same

Question 25.Using the area formula for rectangles and principles about area that we have studied, give a clear and thorough explanation for why the area of the triangle below is 12 ∙ 𝑏 ∙ ℎ square units for the given choices of base 𝑏 and height ℎ

If we enclose the triangle in a rectangle, the rectangle will consist of two copies of the original triangle and two copies of another triangle. The rectangle has area (𝑏 + 𝑎) ∙ ℎ,which is equal to 𝑏 ∙ ℎ + 𝑎 ∙ ℎ by the distributive property.If we put the two small triangles together, they form a rectangle of area 𝑎 ∙ ℎ. If we take this area away from the area of the large rectangle, the remaining area is the area of the two copies of the original triangle combined.Therefore, the area of the two copies of the original triangle is (𝑏 ∙ ℎ + 𝑎 ∙ ℎ) − 𝑎 ∙ ℎ = 𝑏 ∙ ℎ and so the original triangle has half this area, namely area 12 (𝑏 ∙ ℎ)

Question 5: Explain how to determine the sum of the angles A+B+C+D+E in the pentagon below.

If we plot a point inside the pentagon and draw segments from that point to each vertex of the pentagon, we will be forming 5 triangles. The angles of each triangle add to 180 degrees and since we have 5 of them the angles of all the triangles add to 5 x 180=900 degrees. However, this includes the angles formed at the center of the pentagon which are not part of angles A,B,C,D and E; so we need to subtract those angles from 900 degrees. The angles at the center of the pentagon from a full rotation(360 degrees). Therefore A+B+C+D+E=900 degrees -360=540 degrees

Question 32.In the gym there is a round metal pole that is 8 inches in diameter. The gym teacher wants to pole tightly with 1-inch thick rope from the ground up to a height of 6 feet. (The rope will be wound around the pole over and over until it reaches a height of 6 feet.) About how much rope will the gym teacher need? Explain your reasoning

To go around the pole one time, we will need the length of the rope to be equal to the circumference of the pole.𝐶 = 2𝜋(4) = 8𝜋 inches For each time that the rope goes around the pole the height increases by 1 inch. Since we want to get to 6 ft = 72 inches, we need to go around the pole 72 times. This means that we will need 72 times the circumference of the pole of rope to get to 6 feet.72 × 8𝜋 ≈ 1,810 inches = 150 feet and 10 inches

Question 27.Determine the area of the shaded triangle below in two different ways. Explain your reasoning

Way #1. Finding the area of the shaded triangle directly.base = 4 units height = 10 units𝐴 = 12 (4 × 10) = 20 units2Way #2. Using the takeaway strategy.Area of the shaded triangle = Area square - A1 - A2Area of the shaded triangle = (10 × 10) − 12 (10 × 10) − 12 (10 × 6)= 100 − 50 − 30 = 20 units2A 1A2Shaded Area = A1 + A2Shaded Area = 12 (4 × 3) + 12 (5 × 4)= 6 + 10 = 16 cm2

Question 2.Suppose that two lines in a plane meet at a point, as in the figure below. Use the fact that the angle formed by a straight line is 180° to explain why a = c and b = d

Because a and b together make up a straight line a + b = 180°, for the same reason b + c = 180°. So, a = 180° - b and c = 180° - b. Because a and c are both equal to 180° - b, they are equal to each other (a = c). The same argument (with the letters changed) explains why b = d

Question 7: A new giant superstore is being planned somewhere in the vicinity of the Kneebend and Anklescratch, towns which are 10 miles apart. The developers will only say that all the locations they are considering are less than 7 miles from KneeBend and more than 5 miles from Anklecratch. Indicate all the places where the Giant Superstore could be located. Explain your answer.

Because the store needs to be located less than 7 miles fro, Kneebend, it must be inside the circle with a radius of 7 miles with the center at Knneebend. Because the store needs to be more than 5 miles from Anklesctratch, it must be outside the circle with radius of 5 miles with the center at Anklescratch. Therefore, the store could be located anywhere inside the circle with the center at kneebend and outside the circle at Anklescratch.

Question 22.Part b:Use the rectangle to find the result of 2 1/2 × 5 1/2.

By shading a rectangle that is 2 12 cm by 5 12 cm, we can see that we get 10 complete 1 cm2, seven 1/2 cm2, and one 14 cm2; that together is 13 34 cm2.

Question 31.Given that the circumference of a circle of radius 𝑟 units is 2𝜋𝑟 units, explain how to subdivide and rearrange a circle of radius 𝑟 units in order to show why the area of this circle is 𝜋𝑟2 square units

If you cut a circle into 8 "pie pieces" and rearrange them, you get a shape that looks something like a rectangle or a parallelogram. If you cut the circle into 16 or 32 pie pieces and rearrange them, you get shapes that look even more like rectangles. If you could keep cutting the circle into more and more "pie pieces," and keep rearranging them as before, you would get shapes that look more and more like a rectangle.The height of the rectangle is the radius r of the circle. To determine the width of the rectangle (in the horizontal direction), notice that in the rearranged circles, half of the pie pieces point up and half point down. The circumference of the circle is divided equally between the top and bottom sides of the rectangle. The circumference of the circle is 2πr; therefore,the width (in the horizontal direction) of the rectangle is half as much, which is πr. So, the rectangle is r units by πr units, and therefore has area 𝜋𝑟 ∙ 𝑟 = 𝜋𝑟2 squared square units. Since the rectangle is basically a cut-up and rearranged circle of radius r,the area of the rectangle ought to be equal to the area of the circle. Therefore, it makes sense that the area of the circle is also 𝜋𝑟2.

Question 14: Jenny wants to know what it means when we say that a tank is 284 cubic feet. What can you tell Jenny?

If you had a 1-feet-by-1-feet-by-1-feet cube that could be filled with water, then you can fill the tank by pouring the water of such cube 284 times

Question 10: Explain in detail how the sets of right angles, equilateral triangles, and isosceles triangles are related using our definition of these shapes. Make a diagram to show how they are related.

Isosceles triangles have at least 2 sides of the same length. All the sides if an equilateral triangle are the same length. So, equilateral triangles are Isosceles triangles. Right triangles have a right angle and the opposite side to the right angle is called the hypotenuse, which is longer than the other two sides of the triangle. however, these two sides can be the same size making it an isosceles triangle.. So some right triangles are isosceles triangles.

Question 20.When we say that a shape has an area of 15 square centimeters, what does that mean?

It means that the shape can be covered, without gaps or overlaps, with a total of fifteen 1-cm-by-1-cm squares, allowing for squares to be cut apart and pieces to be moved if necessary.

Question 16: When Joe was asked to draw a shape that has an area of 3 square centimeters, he drew a 3 cm by 3cm square. Is Joe right or not? Explain

Joe is not right. A 3 cm by 3 cm square would have an area of 9 cm^2, since it will contain 3 rows of 3one-by-one square cm. A shape that has an area of 3 square centimeters can be represented by any shape that encloses 3 one-by-one square cm

Question 17: Students sometimes say, "area is length times width." Explain why this statement is not fully accurate

Not every shape is a rectangle. For shapes that aren't rectangles, the area is not just "length timeswidth". If something has an area of A square inches, that means that it can be thought of as made from A squares, each 1-inch-by-1-inch

Question 34.Using the figure below, explain why 𝑎2 + 𝑏2 = 𝑐2, where 𝑎 and 𝑏 are the lengths of the short sides of a right triangle, and 𝑐 is the length of the triangle's hypotenuse. (You may assume that all shapes that look like squares really are squares, and that in each drawing, all four triangles with side lengths𝑎, 𝑏, 𝑐 are identical right triangles.)

One way to prove that 𝑎2 + 𝑏2 = 𝑐2 is to imagine taking away the 4 triangles in each large square.Both of the large squares in this figure have sides of length 𝑎 + 𝑏 so both large squares have the same area. Hence, according to the moving and additivity principles, if we remove the 4 copies of the right triangle from each large square, the remaining areas will still be equal. From the square on the left, two smaller squares remain, one with sides of length a and one with sides of length b. Thus,the remaining area on the left is 𝑎2 + 𝑏2. From the square on the right, a single square with sides of length c remains. Therefore, the remaining area on the right is𝑐2. The remaining area on the left is equal to the remaining area on the right, therefore 𝑎2 + 𝑏2 = 𝑐2.

Question 18: Describe one-dimensional, two-dimensional, and three-dimensional parts or aspects of a bottle.What are practical reasons for wanting to know the sizes of these parts or aspects of the bottle?

One-dimensional - the height of the bottle measured in inches or cm, to see if its fits in a particulars pace.Two-dimensional - the area around the bottle measured in square inches or cm2, to determine the amount of paint needed to paint the bottle.Three-dimensional - the volume of liquid that the bottle can hold measured in oz or milliliters, to determine how much liquid the bottle can hold.

Question 28.Determine the area of the shaded shapes in the following figures using only the principles we have studied about area, the area formula for rectangles, and the area formula for triangles (do not use any other area formulas). Explain your reasoning

Shaded Area = A1 + A2Shaded Area = 12 (4 × 3) + 12 (5 × 4)= 6 + 10 = 16 cm2

Question 13: Suppose you use geometry software to construct two circles with centers A and B, in such a way that the circles will always have the same radius, no matter how you move them. Suppose that the two circles meet at points C and D and suppose that you construct line segments to make a quadrilateral ACBD (by connecting A to C, C to B, B to D, and D to A), as shown in the figure below.What kind of special quadrilateral must ACBD be, no matter how you move the points in your construction (as long as the circles still meet at two points)? Explain your answer clearly and in detail, as if you were explaining to someone who was just learning about the geometric concepts involved

Since the two circles have the same radius, the segments AC, AD, BC, and BD (which are radii of the circles) have the same length. This segments form a quadrilateral, with all sides with the same length (the radius of the circle). Therefore, the quadrilateral formed would be a rhombus, no matter how the points are moved in the construction (as long as the circles still meet at two points).

Question 23.Use only the formula for areas of rectangles and the moving and additivity principles about area to determine the area of the shaded shape in the figure below. Explain your method

The area of the whole rectangle (including the missing triangular corners) is 9 × 4 = 36 cm2. Now we need to remove the area ofthe corner triangles, notice that these two triangles together form a rectangle with dimensions 2-cm-by-3-cm having an area of 2 × 3 = 6 cm2. The shaded area is the difference between the area of the square and the area of the triangles. Therefore, the shaded area is 36 - 6 = 30 cm2.

Question 36. What is the longest pole that can fit in a box that is 4 feet wide, 3 feet deep, and 5 feet tall? Explain.

The longest pole fits from the lower front left corner to the upper back right corner. The pole forms a right triangle in which the pole is the hypotenuse, one leg is the height of the box, and the other leg is the diagonal on the bottom of the box.First, we need to find the length of the diagonal on the bottom of the box to then find the length of the pole.To find the length of the diagonal on the bottom of the box we can use the fact that this diagonal is the hypotenuse of the right triangle formed by the diagonal and the width and depth of the box.The longest pole that can fit in the box has a length of 7.07 ft.

Question 11: Using a compass and a straightedge, construct an equilateral triangle. Explain how you know that the triangle is equilateral without measuring it.

The method shown produces an equilateral triangle because of the way it uses circles. Remember that a circle consists of all the points that are the same fixed distance away from the center point. The circle drawn in step 2 consists of all points that are the same distance from A as B is; since C is on this circle, the distance from C to A is the same as the distance from B to A. The circle drawn in step 3 consists of all points that are the same distance from B as A is; since C is also on this circle, the distance from C to B is the same as the distance from A to B. Therefore, the three line segments AB, AC, and BC all have the same length, and the triangle ABC is an equilateral triangle.

Question 24.Show two ways to use the moving and additivity principles to determine the area of the triangle below: one primitive way (that relies directly on what area means) and one more advanced way(that can be generalized to explain the triangle area formula. DO NOT use the triangle area formula). Explain each method

Way 1: Subdivide the triangle into three pieces. Imagine rotating two of those pieces down to form a 2-unit-by-6-unit rectangle as on the right.Since the rectangle was formed by moving portions of the triangle and recombining them without overlapping, by the moving and additivity principles, the original triangle has the same area as the 2-unit-by-6-unit rectangle—namely, 12 square units.Way 2: Subdivide the triangle into two smaller triangles that have areas A and B. Imagine making copies of the two triangles, rotating them,and attaching them to the original triangle to form a 4-unit-by-6-unit rectangle. Now, the area of the triangle is half the area of the rectangle.12 ∙ (4 × 6) = 12 square units. Notice that this is 12 ∙ (base × height)

Question 22.Part a: Explain how to see the large rectangle below as decomposed into 2 12 groups with 5 12 squares in each group, so as to describe the area of the rectangle as 2 12 × 5 12 cm^2.

We can consider the rows to be the groups (which we will only use two and a half of them) and each rows containing 5 12 squares the groups(meaning that we will only use five and a half of the columns).

Question 29.A kite is a quadrilateral that has two pairs of adjacent sides of the same length, as shown below.Find and explain an area formula for kites. Explain what information you need to know to be able to find the area of a kite and give the formula for the area of the kite in terms of that information

We need to know the length of the diagonals of the kite (𝑎 and 𝑏)to find its area.

Question 15: Which of the following describe the same volume, or mean the same as, or are the correct way to read 2 in^3?

a 2-inch by 2-inch by 2-inch cube (this is 8 in^3)• 2 inches cubed• 2 cubic inches• 2 in x 2 in x 2 in (this is 8 in^3

Question 9: For each of the following pairs of sets and shapes, write two sentences about how they are related and draw a diagram that shows how the sets are related.

a) All squares are rhombuses. Not all rhombuses are squares. b) All rectangles are parallelograms. Not all parallelograms are rectangles. c) All parallelograms are trapezoids. Not all trapezoids are parallelograms.

Question 6: Which of the following provides a correct definition from the term circle?

a) You need all points to be a circle not collection b) This will be only 3 points c) This is s sphere d) Correct(All the points in a plane that are one fixed distance away from a point.)

Question 21.You have a 3-foot-by-4-foot rectangular rug in your classroom. You also have a bunch of square foot tiles and some tape measures. a. What is the most primitive way for your students to determine the area of the rug?

a.To cover the cover the rug snugly with the square foot tiles and count that it takes 12 tiles

Question 21.What is a less primitive way for your students to determine the area of the rug and why does this method work?

b. The tiles can be viewed as organized into equal rows (or columns). Because there are equal groups of squares, we can multiply to find the total number of squares. When the 3-foot-by-4-foot rectangular rug is covered with square foot tiles, there are 4 rows of squares with 3 squares in each row. Therefore, there are 4 ∙ 3 squares covering the rug (without overlaps). Each square has area 1 ft^2; therefore, the total area of the rug is 4 ∙ 3 square feet, which is 12 square feet.