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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.

MINI QUEST GEO 4.1Area from CoordinatesRead the lesson
MINI QUEST GEO 4.2Volume as LayersRead the lesson
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

NOTICE → MARK → REASON

Base and height: a matching pair

The height is the perpendicular distance to the chosen base. This is a separate learning diagram; it does not supply the values for your practice question.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

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).

Choose a horizontal base and its perpendicular height.
Base=5, height=8, so area= $\displaystyle \frac{5 \cdot 8}{2}$ .
The area is 20. The top vertex need not lie over the midpoint of the base.

WORKED EXAMPLE 2

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

Choose a horizontal base and its perpendicular height.
Base=6, height=9, so area= $\displaystyle \frac{6 \cdot 9}{2}$ .
The area is 27. The top vertex need not lie over the midpoint of the base.

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

Problem 1
Find the area of the triangle with vertices (0,0), (8,0), and (5,11).
Practise this exact problem →
Problem 2
Find the area of the triangle with vertices (0,0), (9,0), and (6,12).
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Problem 3
Find the area of the triangle with vertices (0,0), (10,0), and (7,13).
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Problem 4
Find the area of the triangle with vertices (0,0), (11,0), and (8,14).
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Problem 5
Find the area of the triangle with vertices (0,0), (12,0), and (9,15).
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Problem 6
Find the area of the triangle with vertices (0,0), (13,0), and (10,16).
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Problem 7New context
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?
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Problem 8New context
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?
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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

TURN IT. LOOK INSIDE. EXPLAIN IT.

Rectangular prism · 3D explorer

Drag to rotate

A worked example you can change. Model values are separate from practice answers. Drag with a mouse or one finger. Use the turn buttons or arrow keys as an alternative.

Explore the solid in motion

$\displaystyle a = 4\, b = 3\, h = 6$

Dashed line: perpendicular height. Your marks use the colors you choose below.

Base length a: 4

Base width b: 3

Perpendicular height h: 6

Show a horizontal cross-sectionShow the two right triangles for the space diagonal

What should you notice?

Every horizontal layer has the same area. Doubling only the height doubles the number of layers and the volume.

Rotating the view changes what you see; it does not change the lengths or volume. Read the labelled dimensions, rather than measuring the screen.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

From the model to the calculation

Find the base area $\displaystyle B = 4 \times 3 \approx 12$ square units.
V = Bh = $\displaystyle \left(4 \times 3\right) \times 6 \approx 72$ cubic units. Stacking equal-area layers explains the multiplication by height.
Check units: base area has square units; multiplying by a length gives cubic units. Rounded values are shown to three decimal places; keep $\displaystyle \pi$ or radicals for exact answers when requested.

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.

Count how many equal-area layers fill the prism.
Base area= $\displaystyle 3 \cdot 4 = 12$ ; multiply by height 3.
Volume=36. Volume uses cubic units; surface area uses square units.

WORKED EXAMPLE 2

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

Count how many equal-area layers fill the prism.
Base area= $\displaystyle 4 \cdot 5 = 20$ ; multiply by height 4.
Volume=80. Volume uses cubic units; surface area uses square units.

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

Problem 1
Find the volume of a rectangular prism with edge lengths 6, 7, and 2.
Practise this exact problem →
Problem 2
Find the volume of a rectangular prism with edge lengths 7, 8, and 3.
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Problem 3
Find the volume of a rectangular prism with edge lengths 8, 9, and 4.
Practise this exact problem →
Problem 4
Find the volume of a rectangular prism with edge lengths 9, 10, and 5.
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Problem 5
Find the volume of a rectangular prism with edge lengths 10, 11, and 2.
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Problem 6
Find the volume of a rectangular prism with edge lengths 11, 12, and 3.
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Problem 7New context
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?
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Problem 8New context
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?
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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 $\displaystyle \left(x, y\right) \to \left(-y, x\right)$ .

\displaystyle R_{90}\, \left(x, y\right) = \left(-y, x\right)

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 $\displaystyle x^{2} + y^{2}$ stays unchanged. Four quarter-turns return every point to its starting position.

WORKED EXAMPLE 1

Rotate $\displaystyle \left(3, -6\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.

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

WORKED EXAMPLE 2

Rotate $\displaystyle \left(4, -7\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.

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

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

Problem 1
Rotate $\displaystyle \left(6, -9\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.
Practise this exact problem →
Problem 2
Rotate $\displaystyle \left(7, -10\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.
Practise this exact problem →
Problem 3
Rotate $\displaystyle \left(8, -11\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.
Practise this exact problem →
Problem 4
Rotate $\displaystyle \left(9, -12\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.
Practise this exact problem →
Problem 5
Rotate $\displaystyle \left(10, -13\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.
Practise this exact problem →
Problem 6
Rotate $\displaystyle \left(11, -14\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.
Practise this exact problem →
Problem 7New context
A robot map rotates a point (12,-15) by a quarter-turn counterclockwise about (0,0). What is the point’s new horizontal coordinate?
Practise this exact problem →
Problem 8New context
A robot map rotates a point (13,-16) by a quarter-turn counterclockwise about (0,0). What is the point’s new horizontal coordinate?
Practise this exact problem →

Open stage GEO 4.3 in the student workspace →

Browse another level →

Coordinates and Solids

Area from Coordinates

Base and height: a matching pair

Understand the idea

Choose and carry out a method

Check the reasoning

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

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

Common pitfalls

Explain it to yourself

Volume as Layers

Rectangular prism · 3D explorer

From the model to the calculation

Understand the idea

Choose and carry out a method

Check the reasoning

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

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

Common pitfalls

Explain it to yourself

Coordinate Rotations

Understand the idea

Choose and carry out a method

Check the reasoning

Rotate (3,−6) 90∘\displaystyle \left(3, -6\right)\, 90^{\circ}(3,−6)90∘ counterclockwise about the origin. Enter the new x-coordinate.

Rotate (4,−7) 90∘\displaystyle \left(4, -7\right)\, 90^{\circ}(4,−7)90∘ counterclockwise about the origin. Enter the new x-coordinate.

Common pitfalls

Explain it to yourself

Rotate $\displaystyle \left(3, -6\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.

Rotate $\displaystyle \left(4, -7\right)\, 90^{\circ}$ counterclockwise about the origin. Enter the new x-coordinate.