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?

   • (10, 7)
   • (16, 10)
   • (18, 11)
   • (13, 8.5)
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?

   • A point equidistant from both endpoints lies on the perpendicular bisector (converse of the bisector theorem).
   • 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.
3. 3.

   Esme draws equal arcs from A and from B (radius over half of AB). They cross at P and Q. Why does line PQ bisect AB at a right angle?

   • Segment PQ is as long as AB, so it must split AB exactly in half where it crosses AB.
   • P and Q are each equidistant from A and B, so both lie on the perpendicular bisector, which two points fix.
   • The arcs cross at the midpoint of AB itself, so the line PQ passes through that midpoint of AB.
   • The compass makes right angles, so line PQ is perpendicular to AB by the construction itself.
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?

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

   Perpendicular bisectors of AB and BC meet at O in Luz's triangle ABC. Why is O the same distance from A, B and C?

   • O is the midpoint of each side, so it is halfway from each corner of the triangle.
   • OA, OB and OC are perpendicular to the sides, so they must be equal in length.
   • O is where the angle bisectors meet, so it is equally far from the three sides.
   • O is on the bisector of AB, so OA = OB, and on the bisector of BC, so OB = OC.
6. 6.

   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, 2 units from B
   • Between B and C, 2 units from B
   • Between A and B, 3 units from B
   • At B
7. 7.

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

   • A line can bisect a segment without being perpendicular to it.
   • 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.
8. 8.

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

   • Bisecting a 120° angle makes two 60° angles.
   • The bisector of a 90° angle makes a 45° angle with each side.
   • 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.
9. 9.

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

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

    • No: P is the midpoint of AB, because the parts of a ratio are equal in size to each other.
    • No: P is 3/4 of the way from A to B, since PB is 3 of the parts.
    • 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.