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Introduction to Geometry / LEVEL 4 · DIFFICULTY 4/5

Coordinates and Solids

Move between diagrams, coordinates and measurement.

3 stages · 24 practice problems · two 6-question assessment forms

Choose an island to read its lesson.

  1. MINI QUEST GEO 4.1Area from CoordinatesRead the lesson
  2. MINI QUEST GEO 4.2Volume as LayersRead the lesson
  3. MINI QUEST GEO 4.3Coordinate RotationsRead the lesson
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STAGE GEO 4.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Area from Coordinates

Useful preparation: Intersecting Chords

Goal: Understand and apply area from coordinates.

Before you begin: Intersecting Chords

base baside c
The height meets the base at a right angle. In this right triangle, side c is the hypotenuse.

Understand the idea

Coordinates can reveal a horizontal base and a perpendicular vertical height without requiring a drawing to scale. A triangle’s area is half the corresponding rectangle’s area.

area=base·height/2

Choose and carry out a method

Subtract endpoint x-coordinates for a horizontal base. Measure the top vertex’s vertical distance from the base line, then multiply and halve.

Check the reasoning

Sliding the top vertex horizontally does not change the height or area. A slanted side’s length is not generally the height.

WORKED EXAMPLE 1

Find the area of the triangle with vertices (0,0), (5,0), and (2,8).

  1. Choose a horizontal base and its perpendicular height.
  2. Base=5, height=8, so area=5·8/2.
  3. The area is 20. The top vertex need not lie over the midpoint of the base.

20

WORKED EXAMPLE 2

Find the area of the triangle with vertices (0,0), (6,0), and (3,9).

  1. Choose a horizontal base and its perpendicular height.
  2. Base=6, height=9, so area=6·9/2.
  3. The area is 27. The top vertex need not lie over the midpoint of the base.

27

Common pitfalls

Possible mix-up: The height is the slanted side.

Height is the perpendicular distance to the base line.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Why does a horizontal shear preserve this triangle’s area?

Preview the eight practice prompts
  1. Find the area of the triangle with vertices (0,0), (8,0), and (5,11).
  2. Find the area of the triangle with vertices (0,0), (9,0), and (6,12).
  3. Find the area of the triangle with vertices (0,0), (10,0), and (7,13).
  4. Find the area of the triangle with vertices (0,0), (11,0), and (8,14).
  5. Find the area of the triangle with vertices (0,0), (12,0), and (9,15).
  6. Find the area of the triangle with vertices (0,0), (13,0), and (10,16).
  7. On a map measured in meters, a triangular planting bed has vertices (0,0), (14,0), and (11,17). What is its area in square meters? New context
  8. On a map measured in meters, a triangular planting bed has vertices (0,0), (15,0), and (12,18). What is its area in square meters? New context
Open stage GEO 4.1 in the student workspace →

STAGE GEO 4.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Volume as Layers

Useful preparation: Area from Coordinates

Goal: Understand and apply volume as layers.

Before you begin: Area from Coordinates

Understand the idea

A prism can be viewed as congruent layers stacked perpendicular to its base. Base area times height measures how many unit cubes fit, including fractional layers when dimensions are not integers.

V=length·width·height

Choose and carry out a method

Find the base area using two perpendicular dimensions. Multiply by the third perpendicular dimension and use cubic units.

Check the reasoning

Reordering the three dimensions changes the chosen base but not the product. Surface area answers a different question.

WORKED EXAMPLE 1

Find the volume of a rectangular prism with edge lengths 3, 4, and 3.

  1. Count how many equal-area layers fill the prism.
  2. Base area=3·4=12; multiply by height 3.
  3. Volume=36. Volume uses cubic units; surface area uses square units.

36

WORKED EXAMPLE 2

Find the volume of a rectangular prism with edge lengths 4, 5, and 4.

  1. Count how many equal-area layers fill the prism.
  2. Base area=4·5=20; multiply by height 4.
  3. Volume=80. Volume uses cubic units; surface area uses square units.

80

Common pitfalls

Possible mix-up: Add the three dimensions for volume.

Volume counts three-dimensional units and requires their product.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

How does doubling just the height compare with doubling every dimension?

Preview the eight practice prompts
  1. Find the volume of a rectangular prism with edge lengths 6, 7, and 2.
  2. Find the volume of a rectangular prism with edge lengths 7, 8, and 3.
  3. Find the volume of a rectangular prism with edge lengths 8, 9, and 4.
  4. Find the volume of a rectangular prism with edge lengths 9, 10, and 5.
  5. Find the volume of a rectangular prism with edge lengths 10, 11, and 2.
  6. Find the volume of a rectangular prism with edge lengths 11, 12, and 3.
  7. A tank has an internal base 12 dm by 13 dm and water depth 4 dm. Each cubic decimeter is one liter. How many liters does it hold? New context
  8. A tank has an internal base 13 dm by 14 dm and water depth 5 dm. Each cubic decimeter is one liter. How many liters does it hold? New context
Open stage GEO 4.2 in the student workspace →

STAGE GEO 4.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Coordinate Rotations

Useful preparation: Volume as Layers

Goal: Understand and apply coordinate rotations.

Before you begin: Volume as Layers

Understand the idea

A quarter-turn about the origin sends the positive x-axis to the positive y-axis. Following both coordinate directions gives the counterclockwise rule (x,y)→(−y,x).

R₉₀(x,y)=(−y,x)

Choose and carry out a method

Apply the rule in order and keep signs attached to each coordinate. Enter only the component requested in the question.

Check the reasoning

The squared distance x²+y² stays unchanged. Four quarter-turns return every point to its starting position.

WORKED EXAMPLE 1

Rotate (3,-6) 90° counterclockwise about the origin. Enter the new x-coordinate.

  1. A 90-degree counterclockwise rotation sends (x,y) to (-y,x).
  2. (3,-6) becomes (6,3).
  3. The new x-coordinate is 6; the distance from the origin is preserved.

6

WORKED EXAMPLE 2

Rotate (4,-7) 90° counterclockwise about the origin. Enter the new x-coordinate.

  1. A 90-degree counterclockwise rotation sends (x,y) to (-y,x).
  2. (4,-7) becomes (7,4).
  3. The new x-coordinate is 7; the distance from the origin is preserved.

7

Common pitfalls

Possible mix-up: Counterclockwise rotation sends (x,y) to (y,−x).

That rule describes a clockwise quarter-turn.

Possible mix-up: A correct numerical answer alone explains the method.

State the governing relationship and check the conditions described above.

Explain it to yourself

Track one point through all four quarter-turns.

Preview the eight practice prompts
  1. Rotate (6,-9) 90° counterclockwise about the origin. Enter the new x-coordinate.
  2. Rotate (7,-10) 90° counterclockwise about the origin. Enter the new x-coordinate.
  3. Rotate (8,-11) 90° counterclockwise about the origin. Enter the new x-coordinate.
  4. Rotate (9,-12) 90° counterclockwise about the origin. Enter the new x-coordinate.
  5. Rotate (10,-13) 90° counterclockwise about the origin. Enter the new x-coordinate.
  6. Rotate (11,-14) 90° counterclockwise about the origin. Enter the new x-coordinate.
  7. A robot map rotates a point (12,-15) by a quarter-turn counterclockwise about (0,0). What is the point’s new horizontal coordinate? New context
  8. A robot map rotates a point (13,-16) by a quarter-turn counterclockwise about (0,0). What is the point’s new horizontal coordinate? New context
Open stage GEO 4.3 in the student workspace →
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