Ray's New Higher Arithmetic · Arts. 385–388 · Unit 26: Evolution

115. Triangles and the right triangle

Goal: Your student knows the parts and kinds of triangles and finds any side of a right triangle from the other two.

⏱ About 40 minutes · 7 steps

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You'll need

  • The book
  • A slate, small whiteboard or scrap paper
  • A notebook and pencil
  • A card or stiff paper, scissors and a ruler

What's in the book

Applications of square and cube root: triangle, base, altitude, the angles of a triangle (with a card illustration), equilateral triangles, the right triangle and hypothenuse, rules and formulas for finding a side, and examples (answers printed).

The lesson, step by step

  1. Warm-up

    Ask: Draw a triangle. Can you name its base and its altitude?

  2. The idea

    Read:

    A Triangle is a figure which has three sides and three angles; as, ABC, MNP.

    The Altitude of a triangle is the perpendicular distance from the base to the vertex of the angle opposite

    The three angles of a triangle are together equal to 180°, or two right angles.

  3. Try the card

    Follow the book's suggestion:

    Mark the angles of a card or paper triangle, 1, 2, 3; and by two cuts divide it into three parts.

    Fit the three corners together along a straight edge. They make a straight line, 180°.

  4. The right triangle

    Read:

    A Right-angled Triangle is a triangle having one right angle

    The side opposite the right angle is called the hypothenuse

    SayThe square on the hypothenuse equals the squares on the other two sides together. Check with 3, 4, 5: 9 + 16 = 25.

  5. The rules

    Read:

    Add together the squares of the base and perpendicular, and extract the square root of the sum.

    Subtract the square of the given side from the square of the hypothenuse, and extract the square root of the difference;

  6. Guided practice

    Find the length of a ladder reaching 12 ft. into the street, from a window 30 ft. high.

    What is the diagonal, or line joining the opposite corners, of a square whose side is 10 ft.?

  7. Independent practice and wrap-up

    Your student works more examples, including the shortcut across a field (69 and 92 rods), and checks the printed answers. Ask: Why is the diagonal always shorter than going around two sides?

Tip

The book spells it "hypothenuse," an older spelling. Today we write "hypotenuse."

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