Analytic Geometry: Test Review

Key facts

Where marks get lost

• ×Subtracting in the midpoint formula. $\frac{x_2-x_1}{2}$ is half the run (how far to move), not the midpoint's coordinate. For $(2,6)$ and $(8,-2)$ the midpoint is $(5,2)$, not $(3,-4)$. (Midpoint and Length of a Line Segment)
• ×Squaring a negative without brackets: typing $-4^2$ into a calculator gives $-16$, but ${(-4)}^2=16$. A sum of squares in the length formula can never be negative. (Midpoint and Length of a Line Segment)
• ×Splitting the square root: $\sqrt{36+16}\ne \sqrt{36}+\sqrt{16}$. Add the squares first, then take one root: $\sqrt{52}=2\sqrt{13}$, not $10$. (Midpoint and Length of a Line Segment)
• ×Mixing coordinates, such as averaging $x_1$ with $y_1$. Average x with x and y with y. (Midpoint and Length of a Line Segment)
• ×Finding a missing endpoint by taking the midpoint of $A$ and $M$. That gives a point a quarter of the way along. Double the move instead: from $A(-3,4)$ and $M(2,-1)$, $B$ is $(7,-6)$. (Midpoint and Length of a Line Segment)
• ×Reading the constant as the radius: $x^2+y^2=16$ has radius $4$, not $16$. Take the square root. (Equation of a Circle)
• ×Forgetting to square the radius when writing the equation: radius $9$ gives $x^2+y^2=81$, not $x^2+y^2=9$. (Equation of a Circle)
• ×Squaring a negative coordinate as a negative: ${(-3)}^2=9$, so $(-3,4)$ gives $9+16=25$, not $-9+16=7$. (Equation of a Circle)
• ×Giving only one answer for $k^2=64$. Both $8$ and $-8$ work, because a horizontal line usually crosses a circle at two points. (Equation of a Circle)
• ×Squaring a radical radius wrongly: ${(2\sqrt{3})}^2=4\times 3=12$, not $2\times 3=6$. (Equation of a Circle)
• ×Taking only half of "negative reciprocal": the slope perpendicular to $2$ is $-\frac{1}{2}$, not $-2$ (negative only) and not $\frac{1}{2}$ (reciprocal only). (Slopes, Perpendicular Bisectors and Distance to a Line)
• ×Using the segment's own slope through the midpoint. That line is the segment's line, not its bisector. For $A(-3,-1)$ and $B(5,3)$, $y=\frac{1}{2}x+\frac{1}{2}$ is wrong; $y=-2x+3$ is right. (Slopes, Perpendicular Bisectors and Distance to a Line)
• ×Drawing the perpendicular through an endpoint instead of the midpoint. That line is perpendicular to the segment but does not bisect it. (Slopes, Perpendicular Bisectors and Distance to a Line)
• ×Measuring the distance to a line along a vertical or horizontal path. From $P(7,0)$ straight up to $y=2x+1$ is $15$ km; the true (perpendicular) distance is $3\sqrt{5}\approx 6.7$ km. (Slopes, Perpendicular Bisectors and Distance to a Line)
• ×Stopping at the foot of the perpendicular. The intersection point $(1,3)$ is where the shortest path ends; the distance is the length from $P$ to that point. (Slopes, Perpendicular Bisectors and Distance to a Line)
• ×Calling a quadrilateral a rectangle because its opposite sides are parallel. That proves only a parallelogram; you also need one right angle (or equal diagonals). (Verifying Properties of Triangles and Quadrilaterals)
• ×Calling a rhombus a square. Four equal sides make a rhombus; it is a square only if it also has a right angle. (Verifying Properties of Triangles and Quadrilaterals)
• ×Checking only one pair of opposite sides. One parallel pair could belong to a trapezoid; a parallelogram needs both pairs. (Verifying Properties of Triangles and Quadrilaterals)
• ×Trusting the sketch: "it looks like a right angle" is not evidence. Calculate the slopes and check that their product is exactly $-1$. (Verifying Properties of Triangles and Quadrilaterals)
• ×Drawing a median to a vertex or perpendicular to a side. A median ends at the midpoint of the opposite side, which is not usually where the perpendicular (the altitude) lands. (Verifying Properties of Triangles and Quadrilaterals)

Practice test

24 questions from across the unit. Each one shows the lesson it comes from.

Endless drill

Fresh numbers every time, for the calculation questions in this unit.