Coordinate geometry

High School Math 1 · Unit 15 · coordinate-geometry · Teacher edition

coordinate-geometry

Name ______________________________Date ______________

1.
Find the distance between $(-2, 3)$ and $(4, 11)$ .
Answer: ______________
2.
Point $P$ lies on the segment from $A(1, 2)$ to $B(9, 14)$ so that $AP : PB = 1 : 3$ . What is the $y$ -coordinate of $P$ ?
Answer: ______________
3.
A triangle has vertices $(1, 1)$ , $(7, 3)$ , and $(3, 6)$ . What is its area, in square units?
Answer: ______________
4.
What is the midpoint of the segment joining $(-5, 2)$ and $(3, 8)$ ?
1. (A)
  $(-1, 5)$
2. (B)
  $(-2, 10)$
3. (C)
  $(4, 3)$
4. (D)
  $(-4, -3)$
5.
Triangle $ABC$ has vertices $A(1, 2)$ , $B(7, 2)$ , and $C(7, 10)$ . What is the length of the median from $B$ to side $\overline{AC}$ ?
1. (A)
  $\sqrt{13}$
2. (B)
  $4$
3. (C)
  $5$
4. (D)
  $\sqrt{34}$
5. (E)
  $\sqrt{41}$

Answer key — Coordinate geometry

1.
10coordinate-geometry-01
$d = \sqrt{6^2 + 8^2} = \sqrt{100} = \mathbf{10}$ — a $6$ – $8$ – $10$ right triangle. ( $100$ is $d^2$ , not $d$ .)
2.
5coordinate-geometry-02
$P = A + \frac{1}{4}(B - A) = (1 + 2,\ 2 + 3) = (3, 5)$ . The $y$ -coordinate is $\mathbf{5}$ . (The midpoint, $(5, 8)$ , would be the $1 : 1$ point.)
3.
13coordinate-geometry-03
Box method: $30 - 6 - 6 - 5 = \mathbf{13}$ . Shoelace check: $\frac{1}{2}\lvert (1 \cdot 3 - 7 \cdot 1) + (7 \cdot 6 - 3 \cdot 3) + (3 \cdot 1 - 1 \cdot 6) \rvert = \frac{1}{2}\lvert -4 + 33 - 3 \rvert = \frac{26}{2} = 13$ . ✓
4.
(A)
$(-1, 5)$
coordinate-geometry-04
$M = \left(\frac{-5 + 3}{2}, \frac{2 + 8}{2}\right) = \mathbf{(-1, 5)}$ . ( $(-2, 10)$ is the sum without halving; $(4, 3)$ halves the differences instead of the sums; $(-4, -3)$ is the negative of that.)
5.
(C)
$5$
coordinate-geometry-05
Midpoint $M = (4, 6)$ . $BM = \sqrt{3^2 + 4^2} = \sqrt{25} = \mathbf{5}$ . (This is half of $AC = 10$ , as it must be: $B$ is the right angle, so the median to the hypotenuse is half the hypotenuse. $\sqrt{34}$ and $\sqrt{41}$ come from misplacing the midpoint; $4$ is only the vertical leg of the $3$ – $4$ – $5$ triangle.)