CAT 2025 Lesson : Trigonometry & Coordinate - Triangle, Quadrilateral & Circle

If the coordinates of the triangle are (5, 5), (14, 5) and (5, 17)

(i) What are the coordinates of the incentre?
(ii) What are the coordinates of the centroid?
(iii) What is the area of the triangle?

Solution

(i) $(a, b)$ are the co-ordinates of the incentre of △PQR.

$x_1 = 5, x_2= 14, x_3 = 5$ and $y_1 = 5, y_2 = 5, y_3 = 17$.
$a = 15, b = 12$ and $c = 9$

⇒ $(a, b) = \left(\dfrac{ax_1 + bx_2 + cx_3}{a + b + c}, \dfrac{ay_1 + by_2 + cy_3}{a + b + c} \right)$

= $\left(\dfrac{((15 \times 5) + (12 \times 14) + (9 \times 5))}{15 + 12 + 9}, \dfrac{((15 \times 5) + (12 \times 5) + (9 \times 17))}{15 + 12 + 9} \right)$

= $\left(\dfrac{(75 + 168 + 45)}{36}, \dfrac{(75 + 60 + 153)}{36} \right)$

= $\left(\dfrac{288}{36}, \dfrac{288}{36} \right)$

$(a, b) = (8, 8)$

(ii) ($c, d$) are the co-ordinates of the centroid of △PQR. $\left(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3} \right)$

⇒ $(c, d) = \left(\dfrac{x_1 + x_2 + x_3}{3}, \dfrac{y_1 + y_2 + y_3}{3} \right) = \left(\dfrac{5 + 14 + 5}{3}, \dfrac{5 + 5 + 17}{3} \right) = (8, 9)$

(iii) Area of the triangle with co-ordinates P $(5, 5)$, Q $(14, 5)$, R $(5, 17)$

⇒ Area of △PQR = $\dfrac{1}{2} \times [(x_1 y_2 - x_2 y_1) + (x_2 y_3 - x_3 y_2) + (x_3 y_1 - x_1 y_3)]$

= $\dfrac{1}{2} \times [(5 \times 5 - 14 \times 5) + (14 \times 17 - 5 \times 5) + (5 \times 5 - 5 \times 17)]$

= $54$

Answer: (i) $(8, 8)$, (ii) $(8, 9)$, (iii) $54$

A line passing through the centre of a circle intersects the circle at $(2, 11)$ and $(-4, 3)$. What is the equation of the circle?

Solution

The line segment connecting the two points $(2, 11)$ and $(-4, 3)$ is the diameter. The mid-point of this line will be the centre of the circle.

Centre of the circle = Mid-point of the diameter = $\left(\dfrac{2 - 4}{2}, \dfrac{11 + 3}{2} \right) = (-1, 7)$

Length of the diameter = $\sqrt{(2 - (-4))^{2} + (11 + 3)^2} = \sqrt{36 + 64} = 10$

Length of the radius = $\dfrac{10}{2} = 5$

Equation of the circle = $(x - a)^{2} + (y - b)^{2} = r^{2} ⇒ (x + 1)^{2} + (y - 7)^{2} = 25$

Answer: $(x + 1)^{2} + (y - 7)^{2} = 25$

What is the area of the quadrilateral formed by the coordinates $(-2, 3), (5, 5), (-2, -4), (3, -5)$?

Solution

Upon plotting the points, we note that the order in which the points have to be arranged to form a quadrilateral is $(-2, 3), (5, 5), (3, -5), (-2, -4)$. Applying the formula, Area = $\dfrac{1}{2}|(x_1 - x_3)(y_2 - y_4) - (x_2 - x_4)(y_1 - y_3)|$ ⇒ $\dfrac{1}{2}|(-2 - (3))(5 - (-4)) - (5 - (-2))(3 - (-5))|$ ⇒ $\dfrac{1}{2}|- 45 - 56| = \dfrac{101}{2} = 50.5$ Answer: $50.5$