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AQA GCSE · Question 18 · Geometry and Measures
J (0, 12) and K (5, 10) are points on the straight line JKLM.
JK = KL = LM
Work out the coordinates of M.
J (0, 12) and K (5, 10) are points on the straight line JKLM.
JK = KL = LM
Work out the coordinates of M.
How to approach this question
1. **Find the vector from J to K:**
To get from J(0, 12) to K(5, 10), we find the change in x and the change in y.
Change in x = 5 - 0 = +5.
Change in y = 10 - 12 = -2.
So the vector JK is (5, -2).
2. **Use the vector to find the other points:**
Since JK = KL = LM, the step from K to L and from L to M is the same vector (5, -2).
3. **Find the coordinates of L:**
Start at K(5, 10) and add the vector.
L = (5 + 5, 10 - 2) = (10, 8).
4. **Find the coordinates of M:**
Start at L(10, 8) and add the vector again.
M = (10 + 5, 8 - 2) = (15, 6).
Full Answer
(15, 6)
The points J, K, L, and M are on a straight line, and the distances between them are equal (JK = KL = LM). This means that to get from J to K, K to L, and L to M, we perform the exact same "step" or vector translation.
**Step 1: Find the step from J to K.**
- J has coordinates (0, 12).
- K has coordinates (5, 10).
- To get from x=0 to x=5, we add 5. (Change in x is +5).
- To get from y=12 to y=10, we subtract 2. (Change in y is -2).
So, the step to get from one point to the next is "go 5 units right and 2 units down".
**Step 2: Find the coordinates of L.**
We apply this step to the coordinates of K.
- K = (5, 10)
- L = (5 + 5, 10 - 2) = (10, 8).
**Step 3: Find the coordinates of M.**
We apply the same step to the coordinates of L.
- L = (10, 8)
- M = (10 + 5, 8 - 2) = (15, 6).
The coordinates of M are (15, 6).
Common mistakes
✗ Making a sign error when finding the change in coordinates (e.g., saying change in y is +2 instead of -2).\n✗ Only applying the step once to find L instead of twice to find M.\n✗ Adding the coordinates instead of finding the difference.
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