The height of the triangle is approximately [tex]4.35\text{ mm}[/tex]
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 [tex]a'[/tex], [tex]b'[/tex] and [tex]c'[/tex], then the semi-perimeter [tex]S[/tex] of the triangle is represented by the expression,
[tex]S=\frac{a'+b'+c'}{2}[/tex]
The area [tex]A[/tex] of the traingle is formulated below.
[tex]\fbox {\begin\\A=\sqrt{s(s-a')(s-b')(s-c')}\end{minispace}}[/tex]
To calculate the area of the triangle with sides [tex]9 \text{ mm}[/tex] , [tex]6 \text{ mm}[/tex] and [tex]12 \text{ mm}[/tex], first find the semi-perimeter.
[tex]S=\frac{9+6+12}{2}\\S=\frac{27}{2}\\S=13.5 \text{ mm}[/tex]
Now, the area of the triangle is calculated below.
[tex]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[/tex]
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,
[tex]\fbox{\begin\\A= \left[\frac{1}{2}(B)(P)\right]\\\end{minispace}}[/tex]
Now, the area of the same triangle can also be calculated as,
[tex]A=\frac{1}{2}(12)(x)\\A=6x[/tex]
In the above calculations, area of the triangle is calculated in two ways.
Therefore, both the areas can be equated to obtain the altitude [tex]x[/tex].
[tex]6x=26.122\\x=\frac{26.122}{6}\\x=4.35[/tex]
Thus, the height of the triangle is evaluated as [tex]\fbox{4.35 \text{ mm}}[/tex].
Learn more:
1. Prove that AB2+BC2=AC2https://brainly.com/question/1591768
2. Which undefined term is needed to define an angle? https://brainly.com/question/3717797
3. Look at the figure, which trigonometric ratio should you use to find x? https://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
