The point on the X-axis which is equidistant from the points (2, -5) and (-2, 9) is
1
(-7, 0)
2
(0, -7)
3
(7, 0)
4
(0, 7)
Official Solution
Correct Option:
(1)
Step 1: Represent the point on the X-axis.
A point on the X-axis has coordinates of the form , since its -coordinate is .
Step 2: Use the distance formula.
The distance between two points and is given by:
Let the point on the X-axis be . The distances from to and must be equal. Thus:
Step 3: Simplify the equation.
Simplify both sides:
Square both sides to eliminate the square roots:
Expand both sides:
Simplify:
Cancel from both sides:
Rearrange terms:
Step 4: Write the coordinates of the point.
The point on the X-axis is .
Final Answer: The point on the X-axis is , which corresponds to option .
02
PYQ 2020
medium
mathematicsID: ap-polyc
The ratio in which the X-axis divide's the line segment joining the points (4, 6) and (3.-8) is
1
1:2
2
2:3
3
3:4
4
4:5
Official Solution
Correct Option:
(3)
Step 1: Recall the section formula.
If a point divides a line segment joining two points and in the ratio , then the coordinates of the dividing point are given by:
Here, the dividing point lies on the X-axis, so its -coordinate is .
Step 2: Use the condition for the X-axis.
The -coordinate of the dividing point is . Using the section formula for the -coordinate:
Substitute and :
Simplify:
Rearrange:
Step 3: Interpret the result.
The ratio is .
Final Answer: The ratio in which the X-axis divides the line segment is , which corresponds to option .
03
PYQ 2020
medium
mathematicsID: ap-polyc
The mid-point of the line joining the points (4, 5) and (-2, -1) is
1
(1, 3)
2
(3, 1)
3
(1, 2)
4
(2, 1)
Official Solution
Correct Option:
(3)
Step 1: Recall the formula for the midpoint.
The midpoint of a line segment joining two points and is given by:
Step 2: Substitute the coordinates of the given points.
The given points are and . Substituting into the formula:
Step 3: Simplify the calculations.
Final Answer: The midpoint is , which corresponds to option .
04
PYQ 2022
medium
mathematicsID: ap-polyc
In the given figure, O is the centre of the circle and ZAOC = 110°, then ZADC is equal to
1
110°
2
55°
3
70°
4
125°
Official Solution
Correct Option:
(2)
We are given that is the centre of the circle and . In the case of a circle, the angle at the centre of the circle is twice the angle subtended at the circumference by the same chord. This means: Substituting the given value of : Now, solving for :
The correct option is (B):
05
PYQ 2022
medium
mathematicsID: ap-polyc
If the points P(2, 3), Q(5, k) and R(6, 7) are collinear, then the value of k is
1
4
2
3
4
6
Official Solution
Correct Option:
(4)
We are given that the points , , and are collinear. For the points to be collinear, the slope between any two points must be the same. Let's calculate the slope between points and , and the slope between points and . The slope between two points and is given by the formula: 1. Slope between and : 2. Slope between and : Since the points are collinear, the slopes must be equal: Now, solve for :
The correct option is (D):
06
PYQ 2024
easy
mathematicsID: ap-polyc
In the following figure, if DE || BC, then x =
1
2
3
4
Official Solution
Correct Option:
(2)
Since DE is parallel to BC, the BPT states that the ratio of corresponding sides is equal.
Therefore: AD/AB = AE/AC = DE/BC
Substitute the given values: (x + 4) / (x + 4 + x + 3) = (2x - 1) / (2x - 1 + x + 1)
Simplify: (x + 4) / (2x + 7) = (2x - 1) / (3x)
Cross-multiply: 3x(x + 4) = (2x + 7)(2x - 1)
Expand: 3x² + 12x = 4x² + 12x - 7
Solve for x:** 0 = x² - 7 x² = 7 x = √7
So, the correct answer is (2) √7.
07
PYQ 2025
medium
mathematicsID: ap-polyc
and are the midpoints of sides and of a triangle respectively and cm. If , then the length of is
1
cm
2
cm
3
cm
4
cm
Official Solution
Correct Option:
(2)
Step 1: Recall the Midpoint Theorem. The Midpoint Theorem states that the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side. Step 2: Apply the Midpoint Theorem to the given triangle. In triangle , is the midpoint of side , and is the midpoint of side . According to the Midpoint Theorem, the line segment is parallel to the third side , and its length is half the length of . Step 3: Calculate the length of . Given that cm, the length of is: The length of is 5 cm.
08
PYQ 2025
medium
mathematicsID: ap-polyc
If is the midpoint of the line segment joining and , then
1
2
3
4
Official Solution
Correct Option:
(4)
Step 1: Recall the midpoint formula. The midpoint of a line segment joining two points and is given by: Step 2: Apply the midpoint formula to the given points. We are given , , and the midpoint . Using the midpoint formula for the x-coordinate: Using the midpoint formula for the y-coordinate: Step 3: Solve for using the x-coordinate equation. Multiply both sides by 3 to find : Step 4: Verify the y-coordinate. Let's check if the y-coordinate of the midpoint matches: The y-coordinate matches the given midpoint. Thus, the value of is .
About The Mid Point Theorem - AP-POLYCET
The Mid Point Theorem is a vital chapter for AP-POLYCET aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.
By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.
Frequently Asked Questions
Why focus on The Mid Point Theorem PYQs?
Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.
How to best use this analysis?
Review the topic breakdown to see which sub-topics within The Mid Point Theorem carry the most weight. Then, tackle the questions iteratively to solidify your understanding.
