Chapter 11.8, Problem 5PSB

To determine

To find: The area of the figure which is formed by combining two triangles with all sides equal.

Answer to Problem 5PSB

Area of the figure which contains two triangles with equal sides is 32.5.

Explanation of Solution

Given Information:

The quadrilateral has two triangles with equal sides.

The sides of the triangles are:

  $a=5,b=7\text{and}c=10$

Formula used:

Using Heron’s formula,

Area of a triangle $\left(ar\Delta \right)=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

Here, $s=\frac{a+b+c}{2}$ is the semi-perimeter of a triangle

  $a,b\text{and}c$ are the sides of a triangle

Calculation:

The given quadrilateral contains two triangles with equal sides.

The sides of the triangles are:

  $a=5,b=7\text{and}c=10$

  $"c"$ is the diagonal of quadrilateral and also one of the sides of the triangle.

Area of the figure = $2\times$ Area of a triangle

We know that, Area of a triangle $\left(ar\Delta \right)=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

  $"s"$ is the semi-perimeter of the triangle

Now, $s=\frac{a+b+c}{2}$

  $\begin{array}{l}s=\frac{5+7+10}{2}\\ s=\frac{22}{2}\\ s=11\end{array}$

So, $ar\Delta =\sqrt{11\left(11-5\right)\left(11-7\right)\left(11-10\right)}$

  $\begin{array}{l}ar\Delta =\sqrt{11\left(6\right)\left(4\right)\left(1\right)}\\ ar\Delta =\sqrt{11\times 2\times 3\times 2^2}\\ ar\Delta =2\sqrt{66}\end{array}$

So, Area of the figure = $2\times$ Area of a triangle

Area of the figure = $4\sqrt{66}=32.5$

Hence, the area of the figure $\mathrm{32.5.}$