Chapter 7
Right Triangles and Trigonometry
A ladder leans against a wall. How could you describe how high it reaches using measurements taken near the ground? You have used similarity to relate lengths in triangles that share a shape. Right triangles and trigonometry bring a new question into focus: how much can a few lengths and angles tell us about an entire triangle?
What You'll Explore
6 lessons.
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Can a single right triangle contain smaller triangles with the same shape? Investigate the pieces formed by an altitude to its longest side, and ask what similarity might reveal about the lengths in the picture.
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Do the side lengths of a triangle hold enough information to tell whether one angle is right? Investigate a connection between those lengths and consider what the Pythagorean theorem can reveal about a triangle you have not drawn.
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A diagonal of a square and an altitude of an equilateral triangle each create right triangles. Investigate what makes these triangles special and what questions their angles raise about the lengths of their sides.
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What changes, and what stays the same, when a right triangle grows but keeps its angles? Investigate ratios of its side lengths as you explore the ideas behind sine, cosine, and tangent.
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Suppose you know one side and one acute angle of a right triangle: is that enough to find every missing measurement? Explore how the information you start with shapes the search for unknown sides and angles.
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Looking up at a building, what could you measure without reaching its roof? Explore questions about heights and slopes, and consider which details a diagram must capture before you can use trigonometry in a real setting.