Perimeter and area of shapes on the coordinate plane

Answer Key - Grade 10 Math Worksheet

Circle the best answer, or write your answer on the line.

1. 1.

   Luz's tile has corners A(2, -4), B(5, 0), C(-3, 6), D(-6, 2). Which is the most specific shape the slopes and lengths prove it to be?

   • Rectangle: opposite sides are parallel and AB ⊥ BC (slopes 4/3 and -3/4), so every angle is right, but AB ≠ BC
   • Square: all four angles are right angles and all four sides are equal
   • Rhombus: all four sides are equal, with AB = BC = 5, so the tile is a diamond with no right angles
   • Parallelogram: both pairs of opposite sides are parallel and equal in length
2. 2.

   Noah's tile has corners A(0, 0), B(3, 4), C(-1, 7), D(-4, 3). Which is the most specific shape the slopes and lengths prove it to be?

   • Rhombus only: all four sides are 5, but AB is not perpendicular to BC
   • Rectangle: opposite sides are parallel and AB ⊥ BC, so all angles are right angles
   • Square: all four sides are 5 and AB ⊥ BC (slopes 4/3 and -3/4)
   • Parallelogram only: opposite sides are parallel and equal, with no right angle
3. 3.

   Pia's tile has corners A(-4, -2), B(1, -1), C(4, 3), D(-1, 2). Which is the most specific shape the slopes and lengths prove it to be?

   • Trapezoid: AB ∥ DC (slope 1/5), but AD is not parallel to BC, so one pair of sides is parallel
   • Quadrilateral: the four corners are joined by four straight sides, and nothing more is proved by the coordinates
   • Parallelogram only: opposite sides are parallel and equal, but AB ≠ AD and AB is not perpendicular to AD
   • Rectangle: opposite sides are parallel and equal in length, so the tile has right angles at the corners
4. 4.

   Ren wants to prove ABCD is a parallelogram. Which computed fact is enough on its own?

   • The midpoint of AC is (0, 0.5), which lies inside the tile
   • AB = DC = √26, so the opposite sides are the same length
   • AB ∥ DC and AB = DC: both run 5 across and 1 up
   • AB ∥ DC, both with slope 1/5
5. 5.

   Tala has shown ABCD is a parallelogram and now wants to prove it is a rectangle. Which computed fact is enough?

   • AB ∥ DC, both with slope 4/3
   • AB = DC = 5
   • AC = BD: both diagonals have length √125
   • The diagonals AC and BD bisect each other
6. 6.

   Zee has shown ABCD is a parallelogram and now wants to prove it is a rhombus. Which computed fact is enough?

   • AB = DC, and AD = BC
   • AB ∥ DC, both with slope 3/4, so opposite sides run the same way
   • The diagonals AC and BD bisect each other
   • AC ⊥ BD: AC is horizontal and BD is vertical
7. 7.

   Ari wants to prove triangle ABC has a right angle at B. Which computed fact is enough?

   • The slope of AB is 4/3
   • AB + BC > AC, so the three lengths make a triangle
   • AB² + BC² = AC²: 25 + 25 = 50
   • AC is the longest of the three sides
8. 8.

   Bo's bike path runs from A(-6, 2) to B(4, -8). Point P is on AB with AP:PB = 2:3. Where is P? Write it like (7, -3).

   (-2, -2)
9. 9.

   Jun's map path runs from A(-5, -2) to B(7, 6). Point P is on AB with AP:PB = 3:1. Where is P? Write it like (7, -3).

   (4, 4)
10. 10.

    Kira's paper path runs from A(-7, 5) to B(5, -7). Point P is on AB with AP:PB = 1:2. Where is P? Write it like (7, -3).

    (-3, 1)