Think and Reflect (Page 5 - After Exercise Set 1.1)
Q1. What are the standard widths for a room door? Look around your home and in school.
Answer:
The standard width for a room door in residential buildings is typically between 2.5 to 3 feet (approximately 76 to 91 cm). For main entrance doors, it may be wider, around 3 to 4 feet. In schools and public buildings, doors are often wider (e.g., 3 to 4 feet or more) to comply with accessibility standards and accommodate wheelchair users. Students should observe and note the actual widths they find in their own surroundings, comparing them to these standard measurements.
Q2. Are the doors in your school suitable for people in wheelchairs?
Answer:
This is a personal response question that requires students to observe the doors in their school. A door width of at least 32 inches (about 81 cm or 2.67 feet) is generally recommended for wheelchair accessibility according to the Americans with Disabilities Act (ADA) standards. Students should measure or estimate the width of school doors, check if there are ramps or level thresholds, and consider whether the doors open easily. If they find the doors are not suitable, they could suggest possible improvements such as installing wider doors, automatic door openers, or ensuring the doorway is free of obstacles.
Think and Reflect (Page 7 - After Quadrants)
Q1. What is the x-coordinate of a point on the y-axis?
Answer:
The x-coordinate of any point on the y-axis is always 0. This is because the x-coordinate represents the perpendicular distance of the point from the y-axis, and a point lying on the y-axis is exactly at zero distance from it. For example, any point on the y-axis has coordinates (0, y) where y can be any real number.
Q2. Is there a similar generalisation for a point on the x-axis?
Answer:
Yes, the y-coordinate of any point on the x-axis is always 0. This is because the y-coordinate represents the perpendicular distance of the point from the x-axis, and a point lying on the x-axis is exactly at zero distance from it. Any point on the x-axis has coordinates (x, 0) where x can be any real number.
Q3. Does point Q (y, x) ever coincide with point P (x, y)? Justify your answer.
Answer:
Point Q (y, x) coincides with point P (x, y) only when x = y. This is because coordinates are ordered pairs — the first number represents the x-coordinate (distance from y-axis) and the second number represents the y-coordinate (distance from x-axis). For two ordered pairs to represent the same point, their corresponding elements must be equal. If x ≠ y, then (x, y) and (y, x) will represent two different points, except when the first coordinate equals the second coordinate. For example, (3, 3) and (3, 3) represent the same point, but (3, 5) and (5, 3) represent different points.
Q4. If x ≠ y, then (x, y) ≠ (y, x); and (x, y) = (y, x) if and only if x = y. Is this claim true?
Answer:
Yes, the claim is absolutely true. Coordinates are ordered pairs, meaning the order of the numbers is crucial. The first element represents the x-coordinate and1 the second represents the y-coordinate. Two ordered pairs are equal only when both corresponding elements are equal. Therefore, (x, y) = (y, x) implies that x = y (from the first elements) and y = x (from the second elements), which is essentially the same condition. Conversely, if x = y, then both pairs are (x, x) and they represent the same point. This is why we say (x, y) ≠ (y, x) when x ≠ y, and (x, y) = (y, x) if and only if x = y. This concept is fundamental to understanding that coordinates are not interchangeable; changing the order changes the location of the point in the plane.
Think and Reflect (Page 9 - After Distance Formula Introduction)
Q1. In moving from A (3, 4) to D (7, 1), what distance has been covered along the x-axis? What about the distance along the y-axis?
Answer:
The distance covered along the x-axis is the difference in the x-coordinates: x₂ - x₁ = 7 - 3 = 4 units. The distance covered along the y-axis is the difference in the y-coordinates: y₁ - y₂ = 4 - 1 = 3 units (taking the absolute value, as distance is always positive).
Q2. Can these distances help you find the distance AD?
Answer:
Yes, these distances represent the two perpendicular sides of a right-angled triangle. The movement along the x-axis (4 units) and the movement along the y-axis (3 units) form the legs of a right triangle. The direct distance AD is the hypotenuse of this right triangle. Using the Baudhāyana–Pythagoras Theorem, we can calculate AD = √(4² + 3²) = √(16 + 9) = √25 = 5 units. This is the fundamental principle behind the distance formula.
Think and Reflect (Page 11 - After Reflection)
Q1. What has remained the same and what has changed with this reflection?
Answer:
With the reflection of triangle ADM in the y-axis, the following observations can be made:
Remained the same:
- The shape and size of the triangle (all side lengths are preserved).
- The y-coordinates of the points remain unchanged.
- The perpendicular distances from the y-axis (the absolute value of x-coordinates) remain the same, but their signs reverse.
- The distance between any two points is preserved.
Changed:
- The x-coordinates of all points become negative (they are reflected to the other side of the y-axis).
- The position of the triangle in the plane changes from Quadrant I to Quadrant II.
- The orientation of the triangle is reversed (it is a mirror image).
Q2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)?
Answer:
Yes, the observations would be similar with the roles of x and y coordinates reversed. If reflected in the x-axis:
Remained the same:
- The shape, size, and all side lengths of the triangle would be preserved.
- The x-coordinates of the points would remain unchanged.
- The perpendicular distances from the x-axis (the absolute value of y-coordinates) would remain the same, but their signs would reverse.
Changed:
- The y-coordinates of all points would become negative (they are reflected below the x-axis).
- The position of the triangle would change from Quadrant I to Quadrant IV.
- The orientation of the triangle would be reversed (mirror image across the x-axis).
In both cases, reflection preserves distances and the shape of the figure, but changes its position and the signs of one of the coordinates.