The height of the triangle is approximately ![4.35\text{ mm}](https://tex.z-dn.net/?f=4.35%5Ctext%7B%20mm%7D)
Step-by-step explanation:
The area of a triangle can be calculated by using the Heron's formula.
<em>Heron's formula: </em>
Suppose a triangle has sides
,
and
, then the semi-perimeter
of the triangle is represented by the expression,
![S=\frac{a'+b'+c'}{2}](https://tex.z-dn.net/?f=S%3D%5Cfrac%7Ba%27%2Bb%27%2Bc%27%7D%7B2%7D)
The area
of the traingle is formulated below.
![\fbox {\begin\\A=\sqrt{s(s-a')(s-b')(s-c')}\end{minispace}}](https://tex.z-dn.net/?f=%5Cfbox%20%7B%5Cbegin%5C%5CA%3D%5Csqrt%7Bs%28s-a%27%29%28s-b%27%29%28s-c%27%29%7D%5Cend%7Bminispace%7D%7D)
To calculate the area of the triangle with sides
,
and
, first find the semi-perimeter.
![S=\frac{9+6+12}{2}\\S=\frac{27}{2}\\S=13.5 \text{ mm}](https://tex.z-dn.net/?f=S%3D%5Cfrac%7B9%2B6%2B12%7D%7B2%7D%5C%5CS%3D%5Cfrac%7B27%7D%7B2%7D%5C%5CS%3D13.5%20%5Ctext%7B%20mm%7D)
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](https://tex.z-dn.net/?f=A%3D%5Csqrt%7Bs%28s-a%29%28s-b%29%28s-c%29%7D%5C%5CA%3D%5Csqrt%7B13.5%2813.5-9%29%2813.5-6%29%2813.5-12%29%7D%5C%5CA%3D%5Csqrt%7B13.5%20%5Ctimes%204.5%20%5Ctimes%207.5%20%5Ctimes%201.5%7D%5C%5CA%3D%5Csqrt%7B%5Cfrac%7B135%7D%7B10%7D%5Ctimes%5Cfrac%7B45%7D%7B10%7D%5Ctimes%5Cfrac%7B75%7D%7B10%7D%5Ctimes%5Cfrac%7B15%7D%7B10%7D%7D%20%5C%5CA%3D%5Csqrt%7B%5Cfrac%7B%2815%5Ctimes3%5Ctimes3%29%20%5Ctimes%20%2815%5Ctimes3%29%20%5Ctimes%20%2815%5Ctimes5%29%20%5Ctimes15%7D%7B100%5Ctimes100%7D%7D%5C%5CA%3D%5Cfrac%7B15%5Ctimes15%5Ctimes3%5Csqrt%7B15%20%7D%20%7D%7B100%7D%20%5C%5CA%3D2.25%5Ctimes3%5Ctimes3.87%5C%5CA%3D26.122)
Area <em>A</em> of a triangle with a altitude <em>P</em> and one side as base <em>B </em>on which the altitude <em>P</em> is drawn, can be calculated as,
![\fbox{\begin\\A= \left[\frac{1}{2}(B)(P)\right]\\\end{minispace}}](https://tex.z-dn.net/?f=%5Cfbox%7B%5Cbegin%5C%5CA%3D%20%5Cleft%5B%5Cfrac%7B1%7D%7B2%7D%28B%29%28P%29%5Cright%5D%5C%5C%5Cend%7Bminispace%7D%7D)
Now, the area of the same triangle can also be calculated as,
![A=\frac{1}{2}(12)(x)\\A=6x](https://tex.z-dn.net/?f=A%3D%5Cfrac%7B1%7D%7B2%7D%2812%29%28x%29%5C%5CA%3D6x)
In the above calculations, area of the triangle is calculated in two ways.
Therefore, both the areas can be equated to obtain the altitude
.
![6x=26.122\\x=\frac{26.122}{6}\\x=4.35](https://tex.z-dn.net/?f=6x%3D26.122%5C%5Cx%3D%5Cfrac%7B26.122%7D%7B6%7D%5C%5Cx%3D4.35)
Thus, the height of the triangle is evaluated as
.
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