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Triangle Midsegments Exploration

Discover and apply properties of triangle midsegments using points A(2, 2), B(12, 2), and C(4, 8). Find midpoints, slopes, lengths, and angles of triangle segments. Explore Triangle Inequality and side length possibilities. Complete assigned exercises on page 336 (11-26).

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Triangle Midsegments Exploration

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  1. Warm Up Use the points A(2, 2), B(12, 2) and C(4, 8) for Exercises 1–5. 1.Find X and Y, the midpoints of AC and CB. 2. Find XY. 3. Find AB. 4. Find the slope of AB. 5. Find the slope of XY. 6. What is the slope of a line parallel to 3x + 2y = 12?

  2. Objective Prove and use properties of triangle midsegments.

  3. A midsegment of a triangleis a segment that joins the midpoints of two sides of the triangle. Every triangle has three midsegments, which form the midsegment triangle.

  4. The relationship shown in Example 1 is true for the three midsegments of every triangle.

  5. Example 2A: Using the Triangle Midsegment Theorem Find each measure. BD ∆ Midsegment Thm. Substitute 17 for AE. BD = 8.5 Simplify.

  6. Example 2B: Using the Triangle Midsegment Theorem Find each measure. mCBD ∆ Midsegment Thm. mCBD = mBDF Alt. Int. s Thm. mCBD = 26° Substitute 26° for mBDF.

  7. The positions of the longest and shortest sides of a triangle are related to the positions of the largest and smallest angles.

  8. The shortest side is , so the smallest angle is F. The longest side is , so the largest angle is G. Example 2A: Ordering Triangle Side Lengths and Angle Measures Write the angles in order from smallest to largest. The angles from smallest to largest are F, H and G.

  9. The smallest angle is R, so the shortest side is . The largest angle is Q, so the longest side is . The sides from shortest to longest are Example 2B: Ordering Triangle Side Lengths and Angle Measures Write the sides in order from shortest to longest. mR = 180° – (60° + 72°) = 48°

  10. A triangle is formed by three segments, but not every set of three segments can form a triangle.

  11. A certain relationship must exist among the lengths of three segments in order for them to form a triangle.

  12. Example 3A: Applying the Triangle Inequality Theorem Tell whether a triangle can have sides with the given lengths. Explain. 7, 10, 19 No—by the Triangle Inequality Theorem, a triangle cannot have these side lengths.

  13. Example 4: Finding Side Lengths The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.

  14. Assignment • Pg. 336 (11-26)

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