Answer:
(a) scalene, see attachment
(b) acute, see attachment
(c) correct; isosceles
Step-by-step explanation:
(a) The sides of the triangle all have different lengths, so the triangle is scalene. Differences in coordinates between the points of the triangle are (6, 2), (5, 2), and (4, 1), so no two distances between these points can be the same.
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(b) The angle opposite the longest side is clearly an acute angle, so the triangle is an acute triangle.
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(c) Miles and Brad live on the same north/south line. Jose lives at a distance that is halfway between those houses in the north-south direction. Hence the distance to Miles' and Brad's houses must be the same from Jose's house. That means the triangle connecting Miles', Brad's, and Jose's houses will be an isosceles triangle.
The minimum distance is the perpendicular distance. So establish the distance from the origin to the line using the distance formula.
The distance here is: <span><span>d2</span>=(x−0<span>)^2</span>+(y−0<span>)^2
</span> =<span>x^2</span>+<span>y^2
</span></span>
To minimize this function d^2 subject to the constraint, <span>2x+y−10=0
</span>If we substitute, the y-values the distance function can take will be related to the x-values by the line:<span>y=10−2x
</span>You can substitute this in for y in the distance function and take the derivative:
<span>d=sqrt [<span><span><span>x2</span>+(10−2x<span>)^2]
</span></span></span></span>
d′=1/2 (5x2−40x+100)^(−1/2) (10x−40)<span>
</span>Setting the derivative to zero to find optimal x,
<span><span>d′</span>=0→10x−40=0→x=4
</span>
This will be the x-value on the line such that the distance between the origin and line will be EITHER a maximum or minimum (technically, it should be checked afterward).
For x = 4, the corresponding y-value is found from the equation of the line (since we need the corresponding y-value on the line for this x-value).
Then y = 10 - 2(4) = 2.
So the point, P, is (4,2).
Answer:
It's a proportion.
2 is to x as 15 is to 9, or 2/x = 15/9.
Multiply the means and extremes
15x = 18
divide both sides by 15 to get x by itself.
x = 1.2
Step-by-step explanation:
