**COORDINATE GEOMETRY - CLASS 10 CHEAT SHEET**
**1. COORDINATE SYSTEM BASICS**
• Coordinate axes: Two perpendicular number lines (x-axis horizontal, y-axis vertical) divide the plane into 4 quadrants
• x-coordinate (abscissa): Distance of point from y-axis
• y-coordinate (ordinate): Distance of point from x-axis
• Point notation: P(x, y) where x is abscissa and y is ordinate
• Points on axes: Any point on x-axis has form (x, 0); any point on y-axis has form (0, y)
• Origin: O(0, 0) is intersection of both axes
**2. DISTANCE FORMULA - THE CORE CONCEPT**
**Formula Statement:**
For any two points P(x₁, y₁) and Q(x₂, y₂), the distance PQ is:
PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]
**Derivation Insight:**
• Use Pythagoras Theorem: In right triangle formed by points P, Q and a third point
• Horizontal distance = |x₂ - x₁|
• Vertical distance = |y₂ - y₁|
• Hypotenuse (actual distance) = √[horizontal² + vertical²]
**Special Cases:**
**3. DISTANCE FORMULA APPLICATIONS & PROBLEM TYPES**
**Type 1: Direct Distance Calculation**
• Apply formula directly
• Substitute coordinates
• Simplify: Square differences → Add → Take positive square root
**Type 2: Checking if Points Form a Triangle**
• Calculate all 3 sides using distance formula
• Check: Sum of any two sides > third side (triangle inequality)
• If inequality holds for all three pairs → triangle exists
• Name triangle type: Use Pythagoras converse (if a² + b² = c², then right triangle)
**Type 3: Verifying Collinearity (Points on Same Line)**
• Calculate distances between all pairs
• If AB + BC = AC → points A, B, C are collinear (B lies between A and C)
• Alternative: Check slopes are equal (if slope AB = slope BC)
**Type 4: Identifying Quadrilateral Properties**
**Verification Method:**
• Calculate all 4 sides
• Calculate both diagonals
• Compare and identify shape
• Alternative: Calculate 4 sides and 1 diagonal; use Pythagoras theorem
**Type 5: Finding Locus (Relation Between Coordinates)**
• Given condition: Point P(x, y) equidistant from two fixed points
• Set: Distance from P to point A = Distance from P to point B
• Expand using distance formula
• Simplify to get relation in x and y
• Result is equation of perpendicular bisector of segment AB
**4. IMPORTANT THEOREMS & CONVERSES**
**Pythagoras Theorem:** In right triangle with sides a, b and hypotenuse c: a² + b² = c²
**Converse of Pythagoras Theorem:** If in a triangle with sides a, b, c we have a² + b² = c², then the angle opposite to side c is 90° (right angle)
**Application:** To prove angle is right angle → show sum of squares of two sides equals square of third side
**5. COMMON MISTAKES & HOW TO AVOID THEM**
**Mistake 1:** Forgetting to square the differences
• Wrong: PQ = |x₂ - x₁| + |y₂ - y₁|
• Correct: PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]
• Avoid: Always square differences first, add them, then take square root
**Mistake 2:** Taking negative square root
• Wrong: Distance = -5
• Correct: Distance always positive; take only positive square root
**Mistake 3:** Confusing collinearity condition
• Wrong: AB + AC = BC means collinear
• Correct: AB + BC = AC means B is between A and C (collinear)
• Check: Which point gives smallest distance? That point lies between other two
**Mistake 4:** Not checking all triangle inequalities
• Wrong: Only check one pair
• Correct: Sum of ANY two sides must be greater than third
• Three conditions must ALL be satisfied
**Mistake 5:** Mixing up distance and coordinate difference
• Distance is always positive magnitude
• Coordinate difference can be negative (tells direction)
• Use: Distance = √[(difference)²] to always get positive value
**Mistake 6:** Arithmetic errors in expansion
• When expanding (x₂ - x₁)²: Remember full expansion is x₂² - 2x₁x₂ + x₁²
• Check signs carefully (subtraction vs addition)
**6. STEP-BY-STEP SOLUTION APPROACH**
**For Distance Problems:**
Step 1: Identify coordinates of both points: P(x₁, y₁) and Q(x₂, y₂)
Step 2: Write distance formula
Step 3: Substitute values
Step 4: Calculate (x₂ - x₁)² and (y₂ - y₁)² separately
Step 5: Add the squares
Step 6: Take positive square root
Step 7: Simplify (rationalize if needed)
**For Verification Problems (Triangle/Quadrilateral):**
Step 1: Label all vertices
Step 2: Find all sides using distance formula (clearly label each)
Step 3: Find both diagonals
Step 4: Compare results with properties table
Step 5: State conclusion with justification
**For Collinearity:**
Step 1: Calculate distance AB, BC, AC
Step 2: Check if AB + BC = AC or AB + AC = BC or AC + BC = AB
Step 3: One equation must be true for collinearity
Step 4: Also verify that points are distinct (distances > 0)
**7. KEY FORMULAS SUMMARY**
• Distance: PQ = √[(x₂ - x₁)² + (y₂ - y₁)²]
• From origin: OP = √(x² + y²)
• Horizontal line: Distance = |difference in x-coordinates|
• Vertical line: Distance = |difference in y-coordinates|
• Collinearity test: If AB + BC = AC, then collinear
• Right angle test: If a² + b² = c², then angle is 90°
Q1. A student plots points A(3, 0), B(7, 0), C(7, 4), and D(3, 4) on a coordinate plane and claims they form a square. Which property must be verified to confirm this claim is incorrect?
Answer: D — These points form a rectangle (not a square) because while all angles are 90°, opposite sides are equal but adjacent sides are not equal (AB = 4, BC = 4 appears equal, but actually AB=4 and BC=4 make it a square only if verified carefully—the error is that the figure IS actually a square here, so the trap is students must recognize BOTH conditions are needed; the distractor A tests whether students think equal sides alone suffice).
Q2. Two points P(x₁, y₁) and Q(x₂, y₂) lie on a line parallel to the x-axis. Which statement correctly describes the distance between them using the distance formula?
Answer: A — The distance formula always applies; when y₁ = y₂, the formula correctly simplifies to |x₂ – x₁|, showing the formula's generality—option B is incomplete phrasing and D confuses 'parallel to x-axis' with vertical distance.
Q3. Assertion (A): If three points form a right triangle, then the sum of squares of two sides equals the square of the third side. Reason (R): The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of squares of the other two sides. Choose the correct option:
Answer: A — A correctly states the property in general form (which side is which depends on identifying the hypotenuse); R provides the exact theorem; R explains why A is true—the converse also holds, validating the statement.
Q4. A construction engineer claims: 'Any quadrilateral with all four sides equal is a square.' Which of the following best refutes this claim using properties from coordinate geometry?
Answer: A — A rhombus is a valid counterexample: all sides equal but diagonals unequal and angles not all 90°—option B is false (equal sides don't guarantee equal angles), C and D are false geometric claims.
Q5. Assertion (A): The distance of the point (5, –3) from the origin is √34. Reason (R): The distance of any point (x, y) from the origin O(0, 0) is given by OP = √(x² + y²). Choose the correct option:
Answer: A — R is the correct formula (derived from the distance formula with origin as reference); applying it: √(5² + (–3)²) = √(25 + 9) = √34, so A is true and R explains it.
Q6. A student uses the distance formula to find PQ where P(2, 3) and Q(2, 7). She calculates: PQ = √[(2–2)² + (7–3)²] = √[0 + 16] = 4. Which statement about her method is most accurate?
Answer: A — The distance formula is universally applicable; her calculation is correct—option C is true but doesn't make A wrong; option B and D are false misconceptions about formula restrictions.
Q7. Assertion (A): If points A, B, and C satisfy AB + BC = AC, then they are collinear (lie on the same line). Reason (R): The distance formula ensures that the sum of two sides of a triangle equals the third side only when the three vertices lie on a straight line. Choose the correct option:
Answer: A — Both A and R are true; R correctly explains A by invoking the triangle inequality—equality holds only in the degenerate case of collinearity.
Q8. Two points P and Q are at distances d₁ and d₂ from the origin respectively. A student claims: 'The distance PQ is always equal to |d₁ – d₂|.' Is this claim valid?
Answer: C — The relationship PQ = |d₁ – d₂| holds only when O, P, Q are collinear and P, Q are on the same side of O (or opposite sides in specific arrangements); in general, PQ depends on the angle between OP and OQ, not just the distances—option C is the correct conditional statement.
Q9. Assertion (A): The points (0, 0), (1, 1), and (2, 2) do not form a triangle. Reason (R): The sum of two sides of a degenerate triangle equals the third side, which occurs when the distance formula gives a zero sum for the triangle inequality. Choose the correct option:
Answer: C — A is true (the three points are collinear); R's phrasing is convoluted and imprecise—the correct reason is that collinear points fail the strict triangle inequality, not because of a 'zero sum' in the distance formula itself.
Q10. A surveyor measures the coordinates of three landmarks: A(0, 0), B(3, 4), and C(6, 8). She wants to verify if they are collinear using coordinate geometry. Which approach is most reliable?
Answer: A — Option A directly applies the collinearity condition (equality in triangle inequality)—options B is imprecise; C requires computing slope (not introduced in this chapter); D is valid but depends on the area formula (not covered here).
What is the distance formula for two points P(x₁, y₁) and Q(x₂, y₂)?
PQ = √[(x₂−x₁)² + (y₂−y₁)²], which uses Pythagoras theorem applied to coordinate differences.
What is the distance of point P(x, y) from the origin O(0, 0)?
OP = √(x² + y²), which is the distance formula with the origin as reference point.
How do you prove three points are collinear using distances?
Calculate distances AB, BC, and AC; if AB + BC = AC, then points lie on the same line.
What condition must be satisfied for a point P(x, y) to be equidistant from A(x₁, y₁) and B(x₂, y₂)?
AP² = BP², which when expanded and simplified gives a linear equation in x and y.
How do you verify that four points form a square using distance formula?
Check that all four sides are equal AND both diagonals are equal, or check sides equal and one angle is 90°.
What is the relationship between distances in a right triangle verified by distance formula?
If AB² + BC² = AC², then by Pythagoras converse, angle B is 90°, confirming a right angle.
If points A(4, 0) and B(6, 0) lie on the x-axis, what is AB?
AB = |6 − 4| = 2 units, since points on the same axis only have coordinate differences in that direction.
What are the coordinates of a point on the x-axis and on the y-axis?
Points on x-axis have form (x, 0) and points on y-axis have form (0, y).
Why can distance never be negative even if (x₂−x₁) or (y₂−y₁) is negative?
Because we square the differences (negative squared becomes positive) and take only the positive square root.
How does the distance formula help in classifying quadrilaterals?
By computing all side lengths and diagonals, we identify if sides are equal, perpendicular angles exist, and determine if shape is square, rectangle, or rhombus.
Define abscissa and ordinate of a point. Write the coordinates of a point lying on the x-axis and a point lying on the y-axis. [2 marks]
Abscissa is distance from y-axis (x-coordinate); ordinate is distance from x-axis (y-coordinate). Points on x-axis have y = 0; points on y-axis have x = 0.
Show that the points A(1, 1), B(5, 5), and C(9, 9) are collinear. Use the distance formula to justify your answer. [3 marks]
Calculate AB, BC, and AC using distance formula; verify if AB + BC = AC to prove collinearity. AB = 4√2, BC = 4√2, AC = 8√2.
Prove that the points A(0, 0), B(3, 4), C(3, 8), and D(0, 4) form a rectangle. Verify that the diagonals are equal and find the area of the rectangle. [5 marks]
Calculate all four sides (AB = 5, BC = 4, CD = 5, DA = 4) showing opposite sides equal; calculate both diagonals AC = BD = 5 using distance formula proving diagonals are equal; area = length × width using side lengths and perpendicularity verified through distance relationships.
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