The ratio of the 2 legs and hypotenuse in this triangle is 1:1:sqrt2
so by proportion length of one leg is 24 / sqrt2 = 16.971
Area of the triangle = 1/2 * 16.971 * 16.971 = 144 sq ft
Answer:
![\large{\textbf{(x,y) = (-7, -15)}\\}](https://tex.z-dn.net/?f=%5Clarge%7B%5Ctextbf%7B%28x%2Cy%29%20%3D%20%28-7%2C%20-15%29%7D%5C%5C%7D)
Step-by-step explanation:
![\large{\textup{The given equations represent two straight lines.}}\\\\ \large{\textup{To solve them is to find the \textbf{point of intersection} of the two lines.}}\\](https://tex.z-dn.net/?f=%5Clarge%7B%5Ctextup%7BThe%20given%20equations%20represent%20two%20straight%20lines.%7D%7D%5C%5C%5C%5C%20%5Clarge%7B%5Ctextup%7BTo%20solve%20them%20is%20to%20find%20the%20%5Ctextbf%7Bpoint%20of%20intersection%7D%20of%20the%20two%20lines.%7D%7D%5C%5C)
![\begin{align*}\\2x - y &= 1 \hspace{25mm} (1) \\3x - y &= -6 \hspace{24mm} (2) \\\end{align*}](https://tex.z-dn.net/?f=%5Cbegin%7Balign%2A%7D%5C%5C2x%20-%20y%20%26%3D%201%20%20%5Chspace%7B25mm%7D%20%281%29%20%5C%5C3x%20-%20y%20%26%3D%20-6%20%5Chspace%7B24mm%7D%20%282%29%20%5C%5C%5Cend%7Balign%2A%7D)
![\large{ \textup{We start by eliminating one variable.}}\\](https://tex.z-dn.net/?f=%5Clarge%7B%20%5Ctextup%7BWe%20start%20by%20eliminating%20one%20variable.%7D%7D%5C%5C)
![\textup{In this case, $y$ can be eliminated by subtracting (1) \& (2).}}}](https://tex.z-dn.net/?f=%5Ctextup%7BIn%20this%20case%2C%20%24y%24%20can%20be%20eliminated%20by%20subtracting%20%281%29%20%5C%26%20%282%29.%7D%7D%7D)
![\large{\textup{We get}\\}](https://tex.z-dn.net/?f=%5Clarge%7B%5Ctextup%7BWe%20get%7D%5C%5C%7D)
![\begin{align*}2x - y &= 1 \\-3x + y &= 6 \\\implies -x &= 7\\\implies x &= -7\end{align*}](https://tex.z-dn.net/?f=%5Cbegin%7Balign%2A%7D2x%20-%20y%20%26%3D%201%20%20%20%5C%5C-3x%20%2B%20y%20%26%3D%206%20%5C%5C%5Cimplies%20-x%20%26%3D%207%5C%5C%5Cimplies%20x%20%26%3D%20-7%5Cend%7Balign%2A%7D)
![\large{\textup{Substituting $x = -7$ in $(1)$, we get:}$ 2(-7) - 1 = y \\ $$\implies y = -15 $\\\textup{Therefore, $x= -7, y = -15$}\\\textup{Simply put, $(x,y) = (-7,-15) $ is the point of intersection of $(1)$ \& $(2)$}.}](https://tex.z-dn.net/?f=%5Clarge%7B%5Ctextup%7BSubstituting%20%24x%20%3D%20-7%24%20in%20%24%281%29%24%2C%20we%20get%3A%7D%24%202%28-7%29%20-%201%20%3D%20y%20%5C%5C%20%24%24%5Cimplies%20y%20%3D%20-15%20%24%5C%5C%5Ctextup%7BTherefore%2C%20%24x%3D%20-7%2C%20y%20%3D%20-15%24%7D%5C%5C%5Ctextup%7BSimply%20put%2C%20%24%28x%2Cy%29%20%3D%20%28-7%2C-15%29%20%24%20is%20the%20point%20of%20intersection%20of%20%24%281%29%24%20%5C%26%20%24%282%29%24%7D.%7D)
Answer:
![\frac{BD}{AD} = \frac{5}{7}](https://tex.z-dn.net/?f=%20%5Cfrac%7BBD%7D%7BAD%7D%20%3D%20%5Cfrac%7B5%7D%7B7%7D%20)
Step-by-step explanation:
Coordinate of B = 1
Coordinate of D = 6
Coordinate of A = -1
BD = |1 - 6| = 5 units
AD = |-1 - 6| = |-7| = 7 units
![\frac{BD}{AD}](https://tex.z-dn.net/?f=%20%5Cfrac%7BBD%7D%7BAD%7D%20)
Plug in the values
![\frac{BD}{AD} = \frac{5}{7}](https://tex.z-dn.net/?f=%20%5Cfrac%7BBD%7D%7BAD%7D%20%3D%20%5Cfrac%7B5%7D%7B7%7D%20)
Answer:
The equation in standard form is: ![y=2x^2-2x-24](https://tex.z-dn.net/?f=y%3D2x%5E2-2x-24)
Step-by-step explanation:
Since they give you the x-intercepts (the zeros of the quadratic expression) one knows that the binomials: (x-(-3)) and (x-4) must be factors of the quadratic expression.
We can therefore write the equation as:
![y=k\,(x+3)(x-4)](https://tex.z-dn.net/?f=y%3Dk%5C%2C%28x%2B3%29%28x-4%29)
using the binomial factors given above, and a numerical factor "k" that we can determine by using the information that the graph passes through the point (2,-20):
![y=k\,(x+3)(x-4)\\-20=k\,(2+3)(2-4)\\-20=k\,(5)(-2)\\-20=k\,(-10)\\k=\frac{-20}{-10} \\k=2](https://tex.z-dn.net/?f=y%3Dk%5C%2C%28x%2B3%29%28x-4%29%5C%5C-20%3Dk%5C%2C%282%2B3%29%282-4%29%5C%5C-20%3Dk%5C%2C%285%29%28-2%29%5C%5C-20%3Dk%5C%2C%28-10%29%5C%5Ck%3D%5Cfrac%7B-20%7D%7B-10%7D%20%5C%5Ck%3D2)
Then,the equation can be written as:
![y=2\,(x+3)(x-4)\\y=2(x^2-x-12)\\y=2x^2-2x-24](https://tex.z-dn.net/?f=y%3D2%5C%2C%28x%2B3%29%28x-4%29%5C%5Cy%3D2%28x%5E2-x-12%29%5C%5Cy%3D2x%5E2-2x-24)
where we wrote the equation already in standard form
There are only 2 possible outcomes.