Find area of the triangle ABC, given the coordinates of vertices in plane

Since you have obtained the length of each side, using Heron's Formula is a natural way to find the area. Let's consider the approach suomynonA suggested in the comments. Consider the figure below.

triangle_inscribed_in_a_rectangle

We can find the area of $\triangle ABC$ by subtracting the sum of the areas of the three right triangles $ABD$, $ACF$, and $BCE$ from the area of rectangle $ADEF$. I will leave the details of the calculations to you.


Follow-through

As you have said before, the side lengths of $\triangle ABC$ is $AB=AC=5\sqrt{5}$, $BC=\sqrt{10}$, using Heron's formula, we can compute the answer.

Heron's formula states that given side lengths $a,b,c$ of $\triangle ABC$, the area is given $$\sqrt{s(s-a)(s-b)(s-c)}\tag{1}$$ Where $s$ Is the semi perimeter. ($s=\frac {a+b+c}{2}$).

So in your case, we have $$a=5\sqrt{5},b=5\sqrt{5},c=\sqrt{10}\tag{2}$$ The semi perimeter is $$\frac {10\sqrt{5}+\sqrt{10}}{2}\tag{3}$$ and plugging in the values, we have $$\sqrt{\frac {10\sqrt{5}+\sqrt{10}}{2}\left(\frac {10\sqrt{5}+\sqrt{10}}{2}-5\sqrt{5}\right)\left(\frac {10\sqrt{5}+\sqrt{10}}{2}-5\sqrt{5}\right)\left(\frac {10\sqrt{5}+\sqrt{10}}{2}-\sqrt{10}\right)}=\boxed{17.5}\tag{4}$$


Method-1 $$\Delta= \frac12\begin{vmatrix} x_1 & y_1 & 1\\ x_2& y_2 & 1\\ x_3& y_3 & 1 \end{vmatrix}$$ Method-2 This can also be used to find the area of polygon. $$\Delta= \frac12\begin{vmatrix} x_1 & y_1 \\ x_2& y_2 \\ x_3& y_3 \\ x_1 &y_1 \end{vmatrix}=\frac12\left ((x_1y_2+x_2y_3+x_3y_1)-(x_2y_1+x_3y_2+x_1y_3)\right)$$

Tags:

Geometry

Area

Triangles

Analytic Geometry

Related

Determinant of matrix with $A_{ij} = \min (i, j)$ How many natural solutions to $x_1+x_2+x_3+x_4=100$ with $x_1 \neq x_2 \neq x_3 \neq x_4$? Multiplication by One The First Homology Group is the Abelianization of the Fundamental Group. Integral using residue theorem complex analysis Knight and Knaves logic problem Weighted sum of two dice such that the result is a random integer between $0$ and $35$ Intuition behind the Thom Isomorphism. An interesting exercise about converging positive series, involving $\sum_{n\geq 1}a_n^{\frac{n-1}{n}}$ 2 conjectured recursion limits for $e$ and $\pi$. Given finitely many points in a vector space $V$, is there a basis such that the first coordinate of each point is distinct? What is the maximum convergent $x$ in the power tower $x^{x^{x^{x\cdots}}}$?

Recent Posts