G_Coordinate Geometry_8_Lesson 10

Pythagoras Theorem

In a right-angled triangle, the Pythagoras formula helps us find the length of one side if we know the other two. It says that the square of the hypotenuse (h) is equal to the sum of the squares of the perpendicular (p) and the base (b). In short, the formula is:
\(p^2 + b^2= h^2\)
Here, the hypotenuse (h) is the longest side and is always opposite the right angle. This rule works only for right-angled triangles and is very useful for solving real-life problems like finding distances or checking if a triangle is right-angled.

How do we identify hypoteneous , perpendicular and base?
Identifying the hypotenuse, perpendicular, and base in a right-angled triangle always depends on the location of the right angle and the reference angle, which is as below.

1. hypotenuse (h)
   Always the longest side of a right-angled triangle. It is always the side opposite the right angle (90°).
2. perpendicular (p), alos called opposite
   The side opposite the reference angle is called perpendicular
3. base (b), also called adjacent
   The side excluding the perpendicular and hypotenuse is called base.

Distance Formula

In a classroom, 2 friends are seated at the points A and B as shown in Figure below. Find distance between them.

Distance between two points is the length of the line segment that connects the two given points. Distance between two points in coordinate geometry can be calculated by finding the length of the line segment joining the given coordinates.
Distance between two points in coordinate geometry is calculated by the formula
\(\sqrt{(x_2 − x_1)^2 + (y_2 − y_1)^2}\)
where \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the coordinate plane.

BLE Questions

1. Find the distance between \( P(x, y) \) and \( Q(a, b) \).[1U]
2. Write the formula to find the distance between the points \( P(x_1, y_1) \) and \( Q(x_2, y_2) \).[1U]
3. Find the distance between the points \( (k, 0) \) and \( (0, k) \).[1U]
4. What will be the \( y \)-coordinate of any point lying on the \( x \)-axis?[1U]
5. In which quadrant does the point \( (8, -5) \) lie?[1K]
6. In which quadrant does the point \( (-5, 2) \) lie?[1K]
7. Find the distance between the points \( A(-3, -4) \) and \( O(0, 0) \).[1U]
8. Find the distance between the points \( (-a, 0) \) and \( (0, -b) \).[1U]
9. If the distance between the points \( A(0, 9) \) and \( B(x, 0) \) is 15 units, find the value of \( x \).[2U]
10. Find the distance between the points \( P(2, 3) \) and \( Q(5, 7) \).[1U]
11. Find the distance between the points \( (-a, a) \) and \( (2a, 5a) \).[1U]
12. If \( AB \), \( BC \), and \( AC \) are the sides of a right-angled \( \triangle ABC \), then express \( AC \) in terms of \( AB \) and \( BC \).[1U]
13. In the adjoining figure, \( \angle B = 90^\circ \), \( AB = 4 \, \text{cm} \), and \( BC = 3 \, \text{cm} \). Find the length of \( AC \).[2U]
14. The \( x \)-coordinate of a point \( A \), which lies on the \( x \)-axis, is \( -8 \); and \( y \)-coordinate of a point \( B \), which lies on the \( y \)-axis, is 6. Find the length of line segment \( AB \).[1U]
15. In the given figure, what is the distance of the point \( P \) from the origin?[1U]
16. Find the distance from the origin to point \( (-6, -8) \).[2U]
17. Find the distance between the points \( A(4, 5) \) and \( B(-4, -5) \).[2U]
18. If the distance between the points \( P(0, 6) \) and \( Q(a, 0) \) is 6 units, then find the value of \( a \).[2U]
19. If the distance between the points \( A(1, -2) \) and \( B(x, -6) \) is \( \sqrt{32} \), then find the value of \( x \).[2U]
20. Prove that the following given points are collinear: \( (3, -1), (1, 1), (-2, 4) \).[2HA]
21. Prove that the following given points are collinear: \( (5, 1), (3, 2), (1, 3) \).[2HA]
22. Find the length of side \( AB \) from the given right-angled triangle.[2U]
23. Determine whether the numbers 6, 8, 10 are Pythagorean triples or not.[2HA]
24. In alongside figure, find the value of \( x \).[2U]
25. A pencil of 13 cm length is kept inside the cylindrical glass as shown in figure. Find the height of the glass if its diameter is 4 cm.[2U]
26. Find the value of \( x \) in the given figure.[2U]
27. In the adjoining figure, find the value of \( PS \).[2U]
28. Find the length of \( AC \) from the given figure of kite.[2U]