Question

Find the height of the triangle by applying formulas for the area of a triangle and your knowledge about triangles.

Answer 1

The height of the triangle is approximately $4.35\text{ mm}$

Step-by-step explanation:

The area of a triangle can be calculated by using the Heron's formula.

Heron's formula:  

Suppose a triangle has sides $a'$, $b'$ and $c'$, then the semi-perimeter $S$ of the triangle is represented by the expression,

$S=\frac{a'+b'+c'}{2}$

The area $A$ of the traingle is formulated below.

$\fbox {\begin\\A=\sqrt{s(s-a')(s-b')(s-c')}\end{minispace}}$

To calculate the area of the triangle with sides $9 \text{ mm}$ , $6 \text{ mm}$ and $12 \text{ mm}$, first find the semi-perimeter.

$S=\frac{9+6+12}{2}\\S=\frac{27}{2}\\S=13.5 \text{ mm}$

Now, the area of the triangle is calculated below.

$A=\sqrt{s(s-a)(s-b)(s-c)}\\A=\sqrt{13.5(13.5-9)(13.5-6)(13.5-12)}\\A=\sqrt{13.5 \times 4.5 \times 7.5 \times 1.5}\\A=\sqrt{\frac{135}{10}\times\frac{45}{10}\times\frac{75}{10}\times\frac{15}{10}} \\A=\sqrt{\frac{(15\times3\times3) \times (15\times3) \times (15\times5) \times15}{100\times100}}\\A=\frac{15\times15\times3\sqrt{15 } }{100} \\A=2.25\times3\times3.87\\A=26.122$

Area A of a triangle with a altitude P and one side as base B on which the altitude P is drawn, can be calculated as,

$\fbox{\begin\\A= \left[\frac{1}{2}(B)(P)\right]\\\end{minispace}}$

Now, the area of the same triangle can also be calculated as,

$A=\frac{1}{2}(12)(x)\\A=6x$

In the above calculations, area of the triangle is calculated in two ways.

Therefore, both the areas can be equated to obtain the altitude $x$.

$6x=26.122\\x=\frac{26.122}{6}\\x=4.35$

 

Thus, the height of the triangle is evaluated as $\fbox{4.35 \text{ mm}}$.

Learn more:

1. Prove that AB2+BC2=AC2https://brainly.com/question/1591768

2. Which undefined term is needed to define an angle? brainly.com/question/3717797

3. Look at the figure, which trigonometric ratio should you use to find x? brainly.com/question/9880052

Answer Details

Grade: Junior High School

Subject: Mathematics

Chapter: Area of triangle

Keywords: area of triangle, heron's formula, base multiplied by height, base multiplied by perpendicular, base multiplied by altitude, right triangle, altitude corresponding to base, area of right triangle