Chapter 11.1, Problem 35PPS

a.

To determine

To Find: To find the area of triangle using Heron’s formula.

Answer to Problem 35PPS

The area is $\frac{7}{4}\sqrt{39}$ square units.

Explanation of Solution

Given information: Given sides of triangle are 7, 10 and 4.

Formula used: Heron’s formula for area of triangle $=\sqrt{s(s-a)(s-b)(s-c)}$

Calculation:

Since the given sides are 7, 10 and 4

Calculating semi perimeter

  $\begin{array}{l}s=\frac{7+10+4}{2}\\ s=\frac{21}{2}\end{array}$

Now, the area of triangle is

  $\begin{array}{l}=\sqrt{\frac{21}{2}\left(\frac{21}{2}-7\right)\left(\frac{21}{2}-10\right)\left(\frac{21}{2}-4\right)}\\ =\sqrt{\frac{21}{2}\times \frac{7}{2}\times \frac{1}{2}\times \frac{13}{2}}\\ =\frac{7}{4}\sqrt{39}\end{array}$

Hence the area is $\frac{7}{4}\sqrt{39}$ square units.

b.

To determine

To Find: To find the area of triangle and verify the results.

Answer to Problem 35PPS

The area is 30square units.

Explanation of Solution

Given information: Given triangle with sides 5, 12 and 13.

Formula used: Area of triangle $=\frac{1}{2}\times \text{base}\times \text{height}$ and Heron’s formula $=\sqrt{s(s-a)(s-b)(s-c)}$

Calculation:

Since the given sides are 5, 12 and 13.

Calculating semi-perimeter first

  $\begin{array}{l}s=\frac{5+12+13}{2}\\ s=\frac{30}{2}\\ s=15\end{array}$

Now, the area of triangle is

  $\begin{array}{l}=\sqrt{15(15-5)(15-12)(15-13)}\\ =\sqrt{15\times 10\times 3\times 2}\\ =\sqrt{900}\\ =30\end{array}$

Thus, the area is 30square units.

Also, the area can be calculated as

  $\begin{array}{l}=\frac{1}{2}\times 5\times 12\\ =30\end{array}$

Thus, area in this case is also 30square units

Hence verified.