1. 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?
2. Can these distances help you find the distance AD?
- 1. 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?
- 2. Can these distances help you find the distance AD?
Step-by-step solution
Idea: Any move from one point to another can be split into a move parallel to the x-axis and a move parallel to the y-axis. These two moves are at right angles.
1. 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?
- Along the x-axis the x-coordinate changes from 3 to 7: distance = 7 − 3 = 4 units (to the right).½ mark
- Along the y-axis the y-coordinate changes from 4 to 1: distance = 4 − 1 = 3 units (downwards).½ mark
2. Can these distances help you find the distance AD?
- Yes. Go from A straight down 3 units to C (3, 1), then 4 units across to D. AC is vertical and CD is horizontal, so the angle at C is a right angle, and AD is the hypotenuse of triangle ACD.½ mark
- By the Baudhāyana–Pythagoras theorem, AD = √(CD2 + AC2) = √(42 + 32) = √25 = 5 units.½ mark
Check: Count on the grid: from A go 3 squares down and 4 squares right to reach D ✓. 3, 4, 5 is a Pythagorean triple: 9 + 16 = 25 ✓.
Answer to write in the exam
1.
Distance along the x-axis = 7 − 3 = 4 units
Distance along the y-axis = 4 − 1 = 3 units
2.
C (3, 1): AC = 3, CD = 4, ∠ACD = 90°
AD2 = CD2 + AC2 (Baudhāyana–Pythagoras theorem) = 16 + 9 = 25
∴ AD = 5 units
Common mistakes that cost marks
- Writing the y-distance as 1 − 4 = −3 units. A distance is never negative; the shift is 3 units (downwards).
- Thinking AD = 4 + 3 = 7. That is the length of the path A → C → D, not the straight line AD.
- Subtracting the wrong coordinates, e.g. 7 − 4 = 3 (an x-coordinate minus a y-coordinate).
How this can come in the exam
A point moves from (−2, 5) to (4, −3). The distances covered along the x-axis and the y-axis are
- 6 and 8
- 2 and 2
- 6 and 2
- 2 and 8
Show answer
(A) 6 and 8
4 − (−2) = 6 and 5 − (−3) = 8.
Assertion (A): The distance between (−3, 5) and (4, −19) is 25 units.
Reason (R): The horizontal and vertical shifts between two points are the legs of a right-angled triangle whose hypotenuse is the distance between them.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
Shifts 7 and 24; by R the distance is √(49 + 576) = √625 = 25.
Try one yourself
In moving from P (1, 1) to Q (3, 5), what distances are covered along the two axes? Hence find PQ.
Show answer
Along the x-axis 2 units, along the y-axis 4 units. PQ = √(4 + 16) = √20 = 2√5 units.
More questions like this
- What if, x1, x2, y1, y2 take negative values? In the figure, triangle AMD is reflected in the y-axis. What are the coordinates of the images of points A, M, and D?
- 1. What has remained the same and what has changed with this reflection?
2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)? - What are the x-coordinate and y-coordinate of the point of intersection of the two axes?
- Point W has x-coordinate equal to −5. Can you predict the coordinates of point H which is on the line through W parallel to the y-axis? Which quadrants can H lie in?
- Consider the points R (3, 0), A (0, −2), M (−5, −2) and P (−5, 2). If they are joined in the same order, predict: