Answer:
Area = 121 cm²
Step-by-step explanation:
We can find the area of the triangle by first finding the lengths of all sides by using the given information.
Let:
- [tex]a=\texttt{first side}[/tex]
- [tex]b=\texttt{second side}[/tex]
- [tex]c=\texttt{third side}[/tex]
- [tex]d=\texttt{altitude to the third side}[/tex]
Now, we convert the information into mathematics equations:
- "The second side of a triangle is 3cm less than twice as long as the first side." → b = 2a - 3 ... [1]
- "The third side of the triangle is 7cm more than three times as long as the first side." → c = 3a + 7 ... [2]
- "The altitude to the third side of the triangle is 1cm more than twice the first side." → d = 2a + 1 ... [3]
Formula of triangle's area:
[tex]\boxed{Area(A)=\frac{1}{2} \times base\times height}[/tex]
If side c is the base, then the height will be side d. Therefore:
[tex]\bf A=\frac{1}{2} cd\ ...\ [4][/tex]
The last information: "Twice the area of the triangle is 88cm^2 more than the product of the second and third sides." → 2A = bc + 88
If we substitute A, b, c and d with equations [1], [2], [3] and [4], it will become:
[tex]\begin{aligned}2A&=bc+88\\2\left(\frac{1}{2}cd\right)&=bc+88\\cd&=bc+88\\c(d-b)&=88\\(3a+7)(2a+1-(2a-3))&=88\\(3a+7)(4)&=88\\3a+7&=22\\3a&=15\\\bf a&\bf=5\ cm\end{aligned}[/tex]
Now, we need to find out c and d by using the equations [2] and [3]:
[tex]\begin{aligned}\\c&=3a+7\\&=3(5)+7\\&=\bf22\ cm\end{aligned}[/tex]
[tex]\begin{aligned}\\d&=2a+1\\&=2(5)+1\\&=\bf11\ cm\end{aligned}[/tex]
[tex]\begin{aligned}\\A&=\frac{1}{2} cd\\\\&=\frac{1}{2} (22)(11)\\\\&=\bf121\ cm^2\end{aligned}[/tex]
