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

   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?

   • The compass makes right angles, so line PQ is perpendicular to AB by the construction itself.
   • 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.
   • Segment PQ is as long as AB, so it must split AB exactly in half where it crosses AB.
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?

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

   • 180°
   • 32°
   • 64°
   • 128°
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 on the bisector of AB, so OA = OB, and on the bisector of BC, so OB = OC.
   • O is where the angle bisectors meet, so it is equally far from the three sides.
   • OA, OB and OC are perpendicular to the sides, so they must be equal in length.
   • O is the midpoint of each side, so it is halfway from each corner of the triangle.
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 B and C, 2 units from B
   • At B
   • Between A and B, 2 units from B
   • Between A and B, 3 units from B
7. 7.

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

   • Every line through the midpoint of AB is perpendicular to AB.
   • A segment has exactly one midpoint.
   • A line can bisect a segment without being perpendicular to it.
   • If M is the midpoint of AB, then AM = ½ AB.
8. 8.

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

   • If ray BD bisects ∠ABC, then m∠ABD = ½ m∠ABC.
   • Bisecting a 120° angle makes two 60° angles.
   • 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.
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, PA = 5 and PB = 4, so P is closer to B than to A.
   • AB = 8 and PA = PB = 5: P sits 3 above the midpoint, off the segment.
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: 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.
    • Yes: the ratio 1 : 3 means 1 part out of 3, so P is 1/3 of the way from A to B.