Question:

What is the area of the triangle whose vertices are (0,0,0) (3, 4, 0) and (3, 4, 6) ?

Updated On: Jul 7, 2022

12 square units
15 square units
30 square units
36 square units

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The Correct Option is B

Solution and Explanation

Let A (0, 0, 0), B (3, 4, 0) and C (3, 4, 6) Area, $(\Delta ) = \sqrt{(\Delta x^2 + \Delta y^2 + \Delta^2})$ Now, $\Delta x = \frac{1}{2} \begin{vmatrix}y_{1}&z_{1}&1\\ y_{2}&z_{2}&1\\ y_{3}&z_{3}&1\end{vmatrix} = \frac{1}{2} \begin{vmatrix}0&0&1\\ 4 &0&1\\ 4&6&1\end{vmatrix} $ $= \frac{1}{2}\left[1\left(24\right)\right] = 12 $ Similarly, $\Delta y = \frac{1}{2} \begin{vmatrix}0&0&1\\ 0 &3&1\\ 6 &3&1\end{vmatrix} = \frac{1}{2}\left[1\left(-8\right)\right]=-9 $ and $\Delta z = \frac{1}{2}\begin{vmatrix}0&0&1\\ 3&4&1\\ 3&4&1\end{vmatrix} = \frac{1}{2}\left[12-12\right]=0$ Area of $ \Delta = \sqrt{12^{2} + \left(-9\right)^{2} +\left(0\right)^{2}} $ $= \sqrt{144+81} = \sqrt{225} = 15 $ square units

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Notes on Straight lines

Concepts Used:

Straight lines

A straight line is a line having the shortest distance between two points.

A straight line can be represented as an equation in various forms, as show in the image below:

The following are the many forms of the equation of the line that are presented in straight line-

1. Slope – Point Form

Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.

y – y₀ = m (x – x₀)

2. Two – Point Form

Let's look at the line. L crosses between two places. P₁(x₁, y₁) and P₂(x₂, y₂) are general points on L, while P (x, y) is a general point on L. As a result, the three points P₁, P₂, and P are collinear, and it becomes

The slope of P₂P = The slope of P₁P₂, i.e.

$\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}$

Hence, the equation becomes:

y - y₁ =$ \frac{y_2-y_1}{x_2-x_1} (x-x1)$

3. Slope-Intercept Form

Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by

y – c =m( x - 0 )

As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if

y = m x +c