Partha, a software engineer, lives in Jerusalem for his work. He lives in the most convenient area of the city from where bank, hospital, post office and supermarket can be easily accessed. In the graph, the bank is plotted as A(9, 5), hospital as B(-3, -1) and supermarket as C(5, -5) such that A, B, C form a triangle. Based on the above given information, answer the following questions : (i) Find the distance between the bank and the hospital. (ii) In between the bank and the supermarket, there is a post office plotted at E which is their mid-point. Find the coordinates of E. (iii) (a) In between the hospital and the supermarket, there is a bus stop plotted as D, which is their mid-point. If Partha wants to reach the bus stand from the bank, then how much distance does he need to cover ? OR (b) P and Q are two different garment shops lying between the bank and the hospital, such that BP = PQ = QA. If the coordinates of P and Q are (1, a) and (b, 3) respectively, then find the values of 'a' and 'b'.

Step 1: Calculate distance between Bank (A) and Hospital (B)

To find the distance between the bank (A) and the hospital (B), we use the distance formula. The coordinates are $A(9, 5)$ and $B(-3, -1)$. We substitute these values into the distance formula.

Step 2: Calculate coordinates of Post Office (E)

The post office (E) is the midpoint between the bank (A) and the supermarket (C). We use the midpoint formula with coordinates $A(9, 5)$ and $C(5, -5)$.

Step 3: Calculate coordinates of Bus Stop (D) and distance AD

The bus stop (D) is the midpoint between the hospital (B) and the supermarket (C). We first find its coordinates using the midpoint formula for $B(-3, -1)$ and $C(5, -5)$. Then, we calculate the distance Partha needs to cover to reach the bus stop (D) from the bank (A) using the distance formula for $A(9, 5)$ and $D(1, -3)$.

Step 4: Calculate coordinates of P and Q using section formula

Points P and Q divide the line segment AB into three equal parts such that $BP = PQ = QA$. This means P divides AB in the ratio $2:1$ and Q divides AB in the ratio $1:2$. We use the section formula for $A(9, 5)$ and $B(-3, -1)$ to find the coordinates of P and Q.

Step 5: Find values of 'a' and 'b'

By comparing the calculated coordinates of P and Q with the given coordinates $P(1, a)$ and $Q(b, 3)$, we can determine the values of 'a' and 'b'.

More from this chapter

• Q7. If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b. [2 Marks]
• Find the ratio in which the line segment joining the points (-3, 10) and (6, -8) is divided by the point (-1, 6).
• Find a point on the y-axis that is at the same distance from the points (-5, 2) and (9, -2).
• Find the coordinates of the point which divides the line segment joining (4, -3) and (8, 5) in the ratio 3:1 internally.
• Find the relation between x and y if the point (x, y) is equidistant from (7, 1) and (3, 5).
• 1. Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii) (-5, 7), (-1, 3) (iii) (a, b), (-a, -b)
• Find the coordinates of the points of trisection of the line segment joining the points A(2, -2) and B(-7, 4).
• Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis, and find the coordinates of the point of division.

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