Question

$\large{ \underbrace{ \blue{ \boxed{\tt{ QuestioN! }}}}}$
If A = (3,-4) , B = (7,0) and C = (14,-7) are the three consecutive vertices of parallelogram ABCD. Find the slope of the diagonal BD.

Note: Please explain completely, I am confused in the question.
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Answer 1

$\textsf{\large{\underline{Solution}:}}$

At first, see the image attached.

Let us name the parallelogram as - ABCD.

1. Coordinates of A = (3, -4)

2. Coordinates of B = (7, 0)

3. Coordinates of C = (14, -7)

Let us assume that the coordinates of D is (x, y)

Let us name the point where the diagonals bisect as O.

Now, let us find out the coordinates of O.

→ O is the midpoint of AC.

$\rm \longrightarrow O = \bigg(\dfrac{3 + 14}{2} , \dfrac{ - 4 - 7}{2} \bigg)$

$\rm \longrightarrow O = \bigg(\dfrac{17}{2} , \dfrac{ - 11}{2} \bigg) - (i)$

Now, O is also the midpoint of BD as diagonals bisect each other. Therefore:

$\rm \longrightarrow O = \bigg(\dfrac{x + 7}{2} , \dfrac{y + 0}{2} \bigg)$

$\rm \longrightarrow O = \bigg(\dfrac{x + 7}{2} , \dfrac{y}{2} \bigg) - (ii)$

Comparing (i) and (ii), we get:

$\rm \longrightarrow (x,y) =(10, - 11)$

Now, we have the coordinate of D. So, the slope of diagonal BD will be:

$\rm = \dfrac{ \Delta y}{ \Delta x}$

$\rm = \dfrac{0 - ( - 11)}{7 - 10}$

$\rm = \dfrac{ - 11}{3}$

Which is our required answer.

$\textsf{\large{\underline{Learn More}:}}$

1. Section formula.

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the point which divides PQ internally in the ratio m₁ : m₂. Then, the coordinates of R will be:

$\rm\longrightarrow R = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)$

2. Mid-point formula.

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the mid-point of PQ. Then, the coordinates of R will be:

$\rm\longrightarrow R = \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg)$

3. Centroid of a triangle formula.

Centroid of a triangle is the point where the medians of the triangle meet.

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle. Let R(x, y) be the centroid of the triangle. Then, the coordinates of R will be:

$\rm\longrightarrow R = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg)$

Answer 2

Answer:

$\huge{\underline{\mathfrak \red{Answer}}}$

At first, see the image attached.

Let us name the parallelogram as - ABCD.

1. Coordinates of A = (3, -4)

2. Coordinates of B = (7, 0)

3. Coordinates of C = (14, -7)

Let us assume that the coordinates of D is (x, y)

Let us name the point where the diagonals bisect as O.

Now, let us find out the coordinates of O.

→ O is the midpoint of AC.

$\rm \longrightarrow O = \bigg(\dfrac{3 + 14}{2} , \dfrac{ - 4 - 7}{2} \bigg)$

$\rm \longrightarrow O = \bigg(\dfrac{17}{2} , \dfrac{ - 11}{2} \bigg) - (i)$

Now, O is also the midpoint of BD as diagonals bisect each other. Therefore:

$\rm \longrightarrow O = \bigg(\dfrac{x + 7}{2} , \dfrac{y + 0}{2} \bigg)$

$\rm \longrightarrow O = \bigg(\dfrac{x + 7}{2} , \dfrac{y}{2} \bigg) - (ii)$

Comparing (i) and (ii), we get:

$\rm \longrightarrow (x,y) =(10, - 11)$

Now, we have the coordinate of D. So, the slope of diagonal BD will be:

$\rm = \dfrac{ \Delta y}{ \Delta x}$

$\rm = \dfrac{0 - ( - 11)}{7 - 10}$

$\rm = \dfrac{ - 11}{3}$

Which is our required answer.

$\textsf{\large{\underline{Learn More}:}}$

1. Section formula.

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the point which divides PQ internally in the ratio m₁ : m₂. Then, the coordinates of R will be:

$\rm\longrightarrow R = \bigg(\dfrac{m_{1}x_{2}+m_{2}x_{1}}{m_{1}+m_{2}}, \dfrac{m_{1}y_{2}+m_{2}y_{1}}{m_{1}+m_{2}}\bigg)$

2. Mid-point formula.

Let P(x₁, y₁) and Q(x₂, y₂) be two points in the coordinate plane and R(x, y) be the mid-point of PQ. Then, the coordinates of R will be:

$\rm\longrightarrow R = \bigg(\dfrac{x_{1}+x_{2}}{2}, \dfrac{y_{1}+y_{2}}{2}\bigg)$

3. Centroid of a triangle formula.

Centroid of a triangle is the point where the medians of the triangle meet.

Let A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) be the vertices of a triangle. Let R(x, y) be the centroid of the triangle. Then, the coordinates of R will be:

$\rm\longrightarrow R = \bigg(\dfrac{x_{1}+x_{2}+x_{3}}{3}, \dfrac{y_{1}+y_{2}+y_{3}}{3}\bigg)$

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