The second side of a triangle is 3cm less than twice as long as the first side. The third side of the triangle is 7cm more than three times as long as the first side. The altitude to the third side of the triangle is 1cm more than twice the first side. twice the area of the triangle is 88cm^2 More than the product of the second and third sides. What is the area of the triangle?



Answer :

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]

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