Cumulative Standards Review
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The given figure is a parallelogram, so both pairs of opposite sides are parallel.
I
We want to find the coordinates of the point U. As arbitrary placeholders, let's label the coordinates of U (a,b).
We know that the given figure is a parallelogram, so both pairs of opposite sides are parallel. Let's find the slopes of the sides by substituting the coordinates of the points into the Slope Formula.
| Side | Slope Formula | Simplify |
|---|---|---|
| SV | 0-0/x+z-x | 0 |
| VU | b-0/a-(x+z) | b/a-x-z |
| UT | y-b/0-a | - y-b/a |
| TS | 0-y/x-0 | - y/x |
Now we can use logic to find the values of a and b in terms of the given variables. Since SV and UT are parallel, their slopes must be equal. - y-b/a= 0 In order for this fraction to equal 0, the numerator must be 0. y-b=0 ⇔ b= y Now we know that the second coordinate of U is y. Since VU and TS are parallel, their slopes are also equal. Additionally, we can replace the b with y in the slope of VU and move the minus sign to the denominator in the expression on the RHS. b/a-x-z= - y/x ⇔ y/a-x-z=y/- x Because the numerators on both sides of the equation are now the same, these slopes will be equal only if the denominators are equal. a-x-z=- x ⇔ a= z This tells us the first coordinate of U. Finally, we have that the coordinates of point U are ( z,y). This corresponds to answer I.