Midpoints and bisectors of segments and angles

Grade 10 Math Worksheet

Circle the best answer.

1. 1.

   A(−2, 1) and B(28, 16) are points on Ari's map. P lies on AB with AP : PB = 2 : 3. What are P's coordinates?

   • (18, 11)
   • (13, 8.5)
   • (10, 7)
   • (16, 10)
2. 2.

   P is 7 cm from A and 7 cm from B. Bo wants to conclude that P lies on the perpendicular bisector of AB. Which reason works?

   • P is 7 cm from both ends of the segment, so P must be the exact midpoint of segment AB itself.
   • Points on the perpendicular bisector are equidistant from A and B, so P is on it, as the theorem says.
   • P is on the perpendicular bisector because it was drawn through the middle of segment AB on the page.
   • A point equidistant from both endpoints lies on the perpendicular bisector (converse of the bisector theorem).
3. 3.

   In Jun's coordinate proof, A(0, 0), B(8, 2) and C(4, 10). M and N are the midpoints of AB and AC. Which computed fact proves MN ∥ BC?

   • MN = √20 and BC = √80, so MN is exactly half the length of BC.
   • M and N are midpoints, so MN ∥ BC is given before any slope is computed.
   • M(4, 1) and N(2, 5) give slope of MN = −2, equal to the slope of BC.
   • The slope of MN is −2 and the slope of AB is 1/4, so the two lines differ.
4. 4.

   Lines AB and CD cross at P. Ray PE bisects ∠APC and ray PF bisects ∠BPD. m∠APC = 64°. What is m∠EPF for Kira's diagram?

   • 32°
   • 64°
   • 128°
   • 180°
5. 5.

   A, B and C lie on a line in that order, with AB = 10 and BC = 6. Milo marks M, the midpoint of AC. Where is M?

   • Between A and B, 3 units from B
   • Between B and C, 2 units from B
   • Between A and B, 2 units from B
   • At B
6. 6.

   Which statement is false, in Noah's notes on midpoints?

   • A segment has exactly one midpoint.
   • If M is the midpoint of AB, then AM = ½ AB.
   • Every line through the midpoint of AB is perpendicular to AB.
   • A line can bisect a segment without being perpendicular to it.
7. 7.

   Which statement is false, in Pia's notes on angle bisectors?

   • Bisecting a 120° angle makes two 60° angles.
   • If ray BD bisects ∠ABC, then m∠ABD = ½ m∠ABC.
   • A ray that makes a 30° angle with one side of a 90° angle bisects it.
   • The bisector of a 90° angle makes a 45° angle with each side.
8. 8.

   Ren argues: M is the midpoint of AB, so line CM is the perpendicular bisector of AB. What is missing from the argument?

   • A proof that M is the midpoint, since that has not been given in the argument.
   • That CM equals AM, which would make line CM perpendicular to segment AB.
   • A second fact, such as CA = CB or CM ⊥ AB, because a line through M can slant.
   • Nothing is missing: a line through a midpoint is a perpendicular bisector.
9. 9.

   Tala claims: if P is equidistant from A and B, then P is the midpoint of AB. Which example breaks the claim?

   • AB = 8 and PA = PB = 4: P is the midpoint of segment AB.
   • AB = 8 and M is the midpoint, with MA = MB = 4 on segment AB.
   • AB = 8 and PA = PB = 5: P sits 3 above the midpoint, off the segment.
   • AB = 8, PA = 5 and PB = 4, so P is closer to B than to A.
10. 10.

    Zee says: if AP : PB = 1 : 3, then P is 1/3 of the way from A to B. Which verdict is right?

    • Yes: the ratio 1 : 3 means 1 part out of 3, so P is 1/3 of the way from A to B.
    • No: AB has 4 equal parts and AP is 1 of them, so P is 1/4 of the way from A.
    • No: P is 3/4 of the way from A to B, since PB is 3 of the parts.
    • No: P is the midpoint of AB, because the parts of a ratio are equal in size to each other.