SME
Grade 10 Math · Geometry
Coordinate geometry: distance, midpoint, partitioning a segment and coordinate proofs
Use this for “Find the distance between (−3, 2) and (5, 8).”
Chart
From (−3, 2) to (5, 8): legs 8 across and 6 up.
d² = 8² + 6² = 64 + 36 = 100
d
=
√
100
=
10
units
This is an example. Type your own values to start a new problem.
d
=
(
−
(
)
)²
+
(
−
)²
Formulas
d
=
√(
(
x₂
−
x₁
)
2
+
(
y₂
−
y₁
)
2
)
10
=
√((5
−
(−3))²
+
(8 − 2)
2
)
Assumptions
•
The segment AB is the hypotenuse of a right triangle whose legs go straight across and straight up.
•
The legs are x₂ − x₁ and y₂ − y₁, so d = √((x₂ − x₁)² + (y₂ − y₁)²).
•
Squaring makes a negative change positive: the order of the points doesn’t matter.
Step-by-step
WE KNOW
x₁ = −3, y₁ = 2, x₂ = 5, y₂ = 8
FIND
distance AB (d)
Step 1 · Find distance AB (d)
Use d = √((x₂ − x₁)² + (y₂ − y₁)²)
The distance formula: the change across and the change up are the legs of a right triangle.
d
=
√((5
−
(−3))²
+
(8 − 2)
2
)
d
=
√
8
2
+
6
2
d
=
√
64 + 36
d
=
√
100
d
=
10
Check
Put the numbers back in. Both sides should match.
10
=
√((5
−
(−3))²
+
(8 − 2)
2
)
✓
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Area from two sides and the angle between them
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The midpoint of a segment
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More problem types
The midpoint of a segment
Use this for “Find the midpoint of the segment from (−4, 3) to (8, −5)” or “M is the midpoint; find the other end.”
The point that splits a segment in a ratio
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Prove a parallelogram with slopes
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Prove a right triangle with slopes
Use this for “Is the triangle A(−1, 1), B(1, 5), C(5, 3) a right triangle?”
Perimeter and area on the grid
Use this for “Find the perimeter and area of the triangle A(0, 0), B(6, 0), C(3, 4).”