Geometry Answer Note #9

Question 1 Find the image of the point - 1,1 on the line 2x-y+5=0 Answer To find the image of a point on a line, we need to understand the concept of reflections. Reflections are a transformation in which a figure is "flipped" over a line. In this case, the line of reflection is the given line, 2x-y+5=0. To find the image of a point on a line, we use the formula (x,y) → (x', y') where x' and y' are the coordinates of the image point. In this case, we need to find the coordinates of the point (-1,1) after it has been reflected over the line 2x-y+5=0. To do so, we need to first find the slope of the given line. The slope-intercept form of a line is y=mx+b, where m represents the slope. Therefore, rearranging the given equation, we can see that the slope of the line is m=2. Now, we need to find the equation of the perpendicular line passing through the point (1,1). The perpendicular slope is the negative reciprocal of the given line's slope. Therefore, the perpendicular slope is -1/2. We can use the point-slope form of a line, yy1=m(x-x1), to find the equation of the perpendicular line passing through the point (1,1). Substituting the known values, we get y-1=-1/2(x-(-1)). Simplifying, we get y=-1/2x-1/2. Now, we can set these two equations equal to each other to find the point of intersection, which is where the reflection will take place. 2x-y+5=0 and y=-1/2x-1/2 Substituting y in the first equation, we get 2x-(-1/2x-1/2)+5=0 2x+1/2x+1/2+5=0 2.5x+5.5=0 2.5x=-5.5 x=-2.2 Substituting x in the second equation, we get y=-1/2(-2.2)-1/2 y=2.1 Therefore, the point of intersection is (-2.2, 2.1). This is where the reflection will take place. To find the image point, we need to find the distance between the original point (1,1) and the point of intersection (-2.2,2.1) and then multiply by 2 since the line of reflection bisects the segment joining these two points. Using the distance formula, we get the distance between the two points to be √(1.2^2+0.1^2) = √1.45 = 1.2 Therefore, the distance between the original point and the image point is 2.4. Since the line of reflection is parallel to the y-axis, the x-coordinate of the image point remains the same. Therefore, the image point is (-1.4, 1.1). Question 2 what surface in the top view is represented by line D in the right-side view Answer After analyzing the given views, it can be determined that line D in the right-side view represents the top surface of the object being viewed. In order to understand this concept further, let us first define what a top view and a right-side view are. A top view, also known as a plan view or bird's eye view, is a representation of an object or space as seen from above, looking down vertically. This view presents a twodimensional outline of the object, with all dimensions in the horizontal plane. On the other hand, a right-side view is a representation of an object as seen from its right side, with all dimensions in the vertical plane. In this view, the height and width of the object are shown, but not its depth. Thus, by combining these two views, we can determine the shape and dimensions of the object in all three dimensions. In conclusion, line D in the right-side view represents the top surface of the object being viewed. Understanding different views of 3D shapes is crucial in fields such as engineering, architecture, and design. By utilizing the sources mentioned, one can have a better understanding of this concept and excel in related subjects. Question 3 Name the three views that are used to describe the shape and size of the part. Answer The three views commonly used to describe the shape and size of a part are the front view, top view, and side view. These views are essential in technical drawing and engineering design as they provide a comprehensive representation of the object. The front view, also known as the front elevation or elevation view, displays the object as seen from the front. This view usually shows the height and width of the object, as well as any relevant details such as holes, slots, or other features. The top view, also called the plan view or bird's eye view, shows the object as seen from above. This view displays the length and width of the object and is useful in representing the overall shape and layout of the part. The side view, also known as the profile view or side elevation, shows the object as seen from the side. This view is particularly useful in representing the thickness or depth of the object, as well as any relevant details that may not be visible in the front view. These three views, when used together, provide a comprehensive and accurate representation of the shape and size of the part. They are often used in combination with cross-sectional views, which show an internal cut-through of the object, to provide a complete understanding of the part's dimensions and features. Question 4 Triangle ABC is shown below: Triangle ABC. Line passes through points D, B, and E. Given: ΔABC Prove: All three angles of ΔABC add up to 180°. The flowchart with missing reason proves the measures of the interior angles of ΔABC total 180°: Top path, by Construction, line segment DE is parallel to line segment AC. By Alternate Interior Angles, angle EBC is congruent to angle BCA. By Substitution, the sum of the measures of angles BCA, CBA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By space labeled 1, angle DBA is congruent to angle BAC. By Substitution, the sum of the measures of angles BCA, BCA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By Definition of a Straight Angle, the measure of angle EBD equals 180 degrees. By Substitution, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees. Bottom path, by Construction, line segment DE is parallel to line segment AC. By Angle Addition Postulate, the sum of the measures of angles EBC, CBA, and DBA equals the measure of angle EBD. By Substitution, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees. Which reason can be used to fill in the numbered blank space? a Alternate Exterior Angles Theorem b Same-Side Interior Angles c Corresponding Angles Postulate d Alternate Interior Angles Theorem Answer Triangle ABC is a commonly studied geometric shape in many math courses. It is important to understand the properties and relationships between the angles in a triangle, as they often play a role in solving various mathematical problems. In this particular case, we are asked to prove that all three angles of triangle ABC add up to 180 degrees. To prove this, we will use the given flowchart and fill in the missing reason for each path. The flowchart shows that line DE is parallel to line AC, and we will use this fact to prove that the interior angles of triangle ABC add up to 180 degrees. Path 1: By Construction, we know that line segment DE is parallel to line segment AC. Using the Alternate Interior Angles Theorem, we can conclude that angle EBC is congruent to angle BCA. This is because they are both alternate interior angles formed by a transversal (line DE) intersecting two parallel lines. Now, using the Substitution property, we can say that the sum of the measures of angles BCA, CBA, and BAC is equal to the measure of angle EBC, which is 180 degrees. Therefore, the first reason that can be used to fill in the blank space is d, the Alternate Interior Angles Theorem. Path 2: Again, by Construction, we know that line DE is parallel to line AC. Using the Sameside Interior Angles Theorem, we can conclude that angle DBA is congruent to angle BAC. This is because they are both interior angles on the same side of the transversal line DE. Similarly to the previous path, using the Substitution property, we can say that the sum of the measures of angles BCA, BCA, and BAC is equal to the measure of angle DBA, which is also 180 degrees. Therefore, the second reason that can be used to fill in the blank space is b, the Same-Side Interior Angles Theorem. Path 3: As before, by Construction, we know that line DE is parallel to line AC. Using the Corresponding Angles Postulate, we can conclude that angle EBD is congruent to angle BCA. This is because they are both corresponding angles formed by a transversal (line DE) intersecting two parallel lines. Using the Substitution property, we can say that the sum of the measures of angles EBC, CBA, and DBA is equal to the measure of angle EBD, which is 180 degrees. Therefore, the third reason that can be used to fill in the blank space is c, the Corresponding Angles Postulate. Path 4: For the final path, by Construction, we know that line DE is parallel to line AC. Using the Angle Addition Postulate, we can say that the measure of angle EBD, which is 180 degrees, is equal to the sum of the measures of angles EBC, CBA, and DBA. Therefore, the fourth reason that can be used to fill in the blank space is d, the Angle Addition Postulate. In conclusion, we can see that all four possible reasons that could fill in the blank space are related to the fact that line segment DE is parallel to line segment AC. By using different angle relationships and properties, we were able to prove that the sum of the measures of angles BCA, BCA, and BAC is equal to 180 degrees. Thus, this provides evidence for our given statement that all three angles of triangle ABC add up to 180 degrees. Question 5 the base of an isosceles obtuse triangle is 18 centimeters, and the radius of the circle described around it is 15 centimeters. Find the lateral side of the triangle Answer To find the lateral side of an isosceles obtuse triangle, we need to first understand the properties of such a triangle. An isosceles triangle has two sides of equal length, and an obtuse triangle has one angle greater than 90 degrees. To begin solving this problem, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In the case of an isosceles obtuse triangle, we can use this theorem to find the length of the lateral side. We know that the radius of the circle described around the triangle is 15 centimeters, which is also equal to the length of the height of the triangle (the perpendicular distance from the base to the opposite vertex). Using this information, we can draw a perpendicular line from the center of the circle to the base of the triangle, forming two right triangles. Since the radius (15 cm) bisects the base (18 cm) of the triangle, each half of the base measures 9 cm. Therefore, using the Pythagorean theorem, we can find the length of the lateral side by finding the square root of (15^2 - 9^2) = √96 ≈ 9.798 cm. This is the length of one of the equal sides of the isosceles obtuse triangle.