Q:

Find the height of the triangle by applying formulas for the area of a triangle and your knowledge about triangles.This is a triangle. side a has a length of 9mm. side b has a length of 6 mm. side c has a length of 12 mm. The altitude to side c has a length of X mm.

Accepted Solution

A:
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=AC2. Which undefined term is needed to define an angle? . Look at the figure, which trigonometric ratio should you use to find x? DetailsGrade: Junior High SchoolSubject: MathematicsChapter: Area of triangleKeywords: 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