Find the area of the triangle whose vertices are $(3{,}8),(-4{,}2)$ and $(5{,}1)$.

Step 1: Identify the coordinates

First, we identify the coordinates of the three vertices of the triangle. Let's label them as $A$, $B$, and $C$ with their respective $(x, y)$ coordinates.

Step 2: Recall the area formula

The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ can be calculated using this determinant-based formula. The absolute value ensures the area is always positive.

Step 3: Substitute the coordinates

Now, we substitute the coordinates of the vertices into the area formula. We replace $x_1, y_1, x_2, y_2, x_3, y_3$ with their respective numerical values.

Step 4: Calculate the expression

Next, we perform the subtractions inside the parentheses and then the multiplications. This simplifies the expression within the absolute value.

Step 5: Simplify and find the area

Finally, we sum the terms inside the absolute value and multiply by $\frac{1}{2}$ to get the area of the triangle. The area is $30.5$ square units.

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]
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• 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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