Perimeter and area of shapes on the coordinate plane

Answer Key - Grade 10 Math Worksheet - Challenge Me

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

1. Read this first

   Use this

   Proof: 1. Slope AB = 3/4 = slope DC, and slope BC = -4/3 = slope AD. 2. Opposite sides are parallel, so ABCD is a parallelogram. 3. ? 4. So ABCD is a rectangle.

2. 1.

   Luz is writing a coordinate proof that this is a rectangle. One step is missing. Which statement belongs in the blank?

   • AB ≠ BC, so ABCD is not a rhombus
   • Slope AB = slope DC, and slope BC = slope AD
   • Slope AB = slope BC, so ABCD has a right angle
   • (3/4)(-4/3) = -1, so AB ⊥ BC

3. Use this

   Proof: 1. The midpoint of AC is (0, 0), and so is the midpoint of BD. 2. The diagonals bisect each other, so ABCD is a parallelogram. 3. ? 4. So ABCD is a rhombus.

4. 2.

   Milo is writing a coordinate proof that this is a rhombus. One step is missing. Which statement belongs in the blank?

   • AC = BD, since both diagonals have the same length
   • Slope AB = slope DC, since both are 4/3
   • ∠A = ∠C, because opposite angles of a parallelogram are equal
   • AB = BC = 10
5. 3.

   Noah is writing a coordinate proof that this is a right triangle. One step is missing. Which statement belongs in the blank?

   • AC = √50, the longest side
   • AB + BC = AC
   • AB = BC, so the triangle is isosceles
   • AB² + BC² = 25 + 25 = 50 = AC²

6. Use this

   Quadrilateral ABCD with A(-5, -1), B(1, -1), C(5, 3), D(-1, 3). Proof: 1. The midpoint of AC is (0, 1). 2. ? 3. The diagonals bisect each other. 4. So ABCD is a parallelogram.

7. 4.

   Pia is writing a coordinate proof that this is a parallelogram. One step is missing. Which statement belongs in the blank?

   • AC = BD, since both diagonals have the same length
   • The midpoint of AB is (-2, -1)
   • The midpoint of BD is (0, 1), the same point
   • The midpoint of AC is (0, 1)
8. 5.

   Ren says ABCD is a square because AB = BC = CD = DA = 5. Which reply finds the error?

   • Equal sides prove a rhombus. A square also needs a right angle, and AB and BC have slopes 3/4 and -3/4, which do not multiply to -1
   • AB and DC have different slopes, so ABCD is not even a parallelogram, and the proof fails at that step
   • Nothing: a quadrilateral with four equal sides is a square, because a square is defined by its sides alone
   • AB is not 5: the run is 4 and the rise is 3, so AB = 7, and the four sides are not equal after all
9. 6.

   Tala says ABCD is a rectangle because AB = DC and AD = BC. Which reply finds the error?

   • AB and DC are not parallel, so ABCD is not even a parallelogram, and the proof fails at that step
   • Nothing: opposite sides that are equal make a rectangle, because both pairs of opposite sides match in length
   • AD is not 5: the run is 3 and the rise is 4, so AD = 7, and the opposite sides are not equal after all
   • Opposite sides equal prove a parallelogram. A rectangle also needs a right angle, and AB and AD have slopes 1/5 and 4/3, which do not multiply to -1
10. 7.

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

    (-1, 3)
11. 8.

    Esme's rug route runs from A(-8, 4) to B(4, -4). Point P is on AB with BP:PA = 1:3. Where is P? Write it like (7, -3).

    (1, -2)
12. 9.

    Jun's tent route runs from A(-6, -5) to B(6, 7). Point P is 2/3 of the way from A to B. Where is P? Write it like (7, -3).

    (2, 3)
13. 10.

    P is not drawn. Kira's flower route runs from A(-4, -2) to B(4, 2). Point P is on line AB beyond B, with AB:BP = 2:1. Where is P? Write it like (7, -3).

    (8, 4)