Respuesta :
Answer:
The height is 6 cm and the base is 5 cm.
Step-by-step explanation:
Given:
The base of a triangle is 9 cm less than the three times the height. The area is 15 cm².
Now, to find the height and the base of the triangle.
Let the height of the triangle be [tex]x.[/tex]
And then base of the triangle = [tex]3x-9.[/tex]
The area of the triangle = [tex]15\ cm^2.[/tex]
Now, we put formula to get the base and height of the triangle:
[tex]Area=\frac{1}{2} \times base\ \times height\\\\15=\frac{1}{2} \times (3x-9)\times x\\\\15=\frac{3x^2-9x}{2}[/tex]
Multiplying both sides by 2 we get:
[tex]30=3x^2-9x[/tex]
[tex]30=3(x^2-3x)[/tex]
Dividing both sides by 3 we get:
[tex]10=x^2-3x[/tex]
Subtracting both sides by 10 we get:
[tex]0=x^2-3x-10\\\\x^2-3x-10=0[/tex]
Now, solving the quadratic equation:
[tex]x^2-5x+2x-10=0\\\\x(x-5)+2(x-5)=0\\\\(x-5)(x+2)=0\\\\\\x-5=0\ \ \ \ \ x+2=0\\\\x=5\ \ \ \ \ \ \ \ \ \ x=-2[/tex]
Hence, the negative result is not to be taken so, we take the positive result.
Thus, the height is 5 cm.
Now, substituting the value of [tex]x[/tex] to get the base:
[tex]3x-9\\\\=3\times 5-9\\\\=15-9\\\\=6\ cm.[/tex]
Hence, the base is 6cm.
Therefore, the height is 6 cm and the base is 5 cm.
