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Distance and Midpoint

Distance and Midpoint. October 2007. Warm-up. A triangle has vertices R (17, 16), S (1, 4), and T (7, –4). Prove that the triangle is right-angled. Hint: (Draw the triangle and find slopes). Find the length of the horizontal line PR and the vertical line QR.

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Distance and Midpoint

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  1. Distance and Midpoint October 2007

  2. Warm-up • A triangle has vertices R (17, 16), S (1, 4), and T (7, –4). Prove that the triangle is right-angled. • Hint: (Draw the triangle and find slopes)

  3. Find the length of the horizontal line PR and the vertical line QR List the coordinates of P, Q, and R P (1,1) R (5, 1) Q (5, 4) PR = 5 – 1 = 4 (subtract the x coordinates for a horizontal line) QR = 4 – 1 = 3 (subtract the y coordinates for a vertical line) Q P R

  4. Find the distance from P to Q • Find the length of QR and PR • Use these lengths and the Pythagorean Theorem, calculate the length of • and • (PQ)2 = 42 + 32 • (PQ)2 = 25 • PQ = 5 Q P R

  5. Solve for the length of PQ, and show the general formula side by side. • (PQ)2 = (5 – 1)2 + (4 -1)2 • (PQ)2 = 42 + 32 • (PQ)2 = 25 • PQ = 5 (PQ)2 = (x2 - x1)2 + (y2 - y1)2 d2 = (x2 - x1)2 + (y2 - y1)2 Solving for d, the distance formula is obtained Q P R

  6. Find the length of JK using the distance formula K Coordinates of J and K J (-3, -5) and K (5, 12) J

  7. A triangle has vertices R (17, 16), S (1, 4), and T (7, –4). Prove that the triangle is right-angled in a different way than you did in the warm-up

  8. Find the midpoint of line JK Remember the coordinates of J and K J (-3, -5) and K (5, 12) K Find the midpoint of the x coordinates Find the midpoint of the y coordinates J

  9. So the midpoint of JK is: The x coordinate of the midpoint is found by K The y coordinate of the midpoint is found by (1, 3.5) Midpoint Formula J

  10. Try this: • Determine the midpoint, M of the line segment with endpoints A (-2, -3) and B(4, 7) • M (1, 2) • What is the length of the line segment? • √136 = 11.66 . . . .

  11. In a circle • The endpoints of the diametre of a circle are located at (-7,5) and (11, 7). • Find the length of the radius of this circle.

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