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. 1.

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

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

2. Read this first

   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.

3. 2.

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

   • The midpoint of AC is (0, 1)
   • The midpoint of AB is (-2, -1)
   • The midpoint of BD is (0, 1), the same point
   • AC = BD, since both diagonals have the same length
4. 3.

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

   • Nothing: a quadrilateral with four equal sides is a square, because a square is defined by its sides alone
   • 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 is not 5: the run is 4 and the rise is 3, so AB = 7, and the four sides are not equal after all
   • AB and DC have different slopes, so ABCD is not even a parallelogram, and the proof fails at that step
5. 4.

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

   • 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
   • 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
   • Nothing: opposite sides that are equal make a rectangle, because both pairs of opposite sides match in length
   • AB and DC are not parallel, so ABCD is not even a parallelogram, and the proof fails at that step
6. 5.

   Zee says triangle ABC has a right angle at B because AB = BC. Which reply finds the error?

   • Equal sides prove an isosceles triangle. A right angle at B needs AB² + BC² = AC², but 25 + 25 = 50 and AC² = 36
   • AB is not equal to BC, so even the first fact is wrong, and the whole claim falls apart there
   • AC is not 6: the run is 6 and the rise is 0, so AC = 36, and the triangle is not isosceles
   • Nothing: two equal sides next to B make the angle at B a right angle, since the triangle looks square at B
7. 6.

   Ari says the point P with AP:PB = 2:3 is 2/3 of the way from A(-6, 2) to B(9, -3). Which reply finds the error?

   • A and B are in the wrong order, so P must be measured from B and not from A
   • AP:PB = 2:3 puts P 2 of 5 equal parts from A, which is 2/5 of the way, not 2/3
   • Nothing: the first number in the ratio is the fraction of the way, so 2:3 means 2/3 of AB
   • B - A is not (15, -5): the run is 3 and the rise is 1, so P is not 2/3 of the way
8. 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)
9. 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)
10. 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)
11. 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)