Midpoints and bisectors of segments and angles

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   Use this

   Proof card. Given: A, B, C, D and E lie on a line in that order. B is the midpoint of AC and D is the midpoint of CE. Prove: AE = 2BD. Step 1: AB = BC and CD = DE. Step 2: BD = BC + CD. Step 3: AE = AB + BC + CD + DE. Step 4: AE = 2BC + 2CD. Step 5: AE = 2BD.

2. 1.

   Ari fills in a proof, shown on the card. What is the reason for Step 5?

   • Definition of midpoint, since B and D are both midpoints.
   • Factor out 2 and use Step 2: 2BC + 2CD = 2(BC + CD) = 2BD.
   • Substitution of Step 1 into Step 2, because AB = BC.
   • Segment addition postulate, since BD lies inside AE.
3. 2.

   Esme says: if m∠ABD = m∠DBC, then ray BD bisects ∠ABC. Which example shows the claim is false?

   • ∠ABC = 60°, and BD lies inside it, making ∠ABD = ∠DBC = 30° on both sides.
   • ∠ABC = 120°, and BD splits it into two 60° angles that both lie inside ∠ABC.
   • ∠ABC = 90°, and BD lies inside it, making ∠ABD = ∠DBC = 45° on both sides.
   • ∠ABC = 120°, and BD points away from the angle, so ∠ABD = ∠DBC = 120°.
4. 3.

   M is the midpoint of AB on Jun's ribbon. AM = (x² − 4x) cm and MB = (3x − 6) cm, with positive lengths. N is the midpoint of MB. What is AN?

   • 24 cm
   • 18 cm
   • 4.5 cm
   • 6 cm
5. 4.

   Ray BD bisects ∠ABC in Kira's drawing. m∠ABD = (x² − 20)° and m∠DBC = (3x + 8)°, with positive angles. What is m∠ABC?

   • 58°
   • 29°
   • 8°
   • 116°
6. 5.

   Ray BD bisects ∠ABC and ray BE bisects ∠DBC in Luz's drawing. m∠ABC = (14x − 12)° and m∠EBC = (2x + 3)°. What is m∠ABE?

   • 11°
   • 44°
   • 22°
   • 33°
7. 6.

   A, B, C, D lie in order on a line: AB = 6, BC = 4, CD = 12. M and N are the midpoints of AB and CD. What is MN for Milo?

   • 22
   • 9
   • 11
   • 13
8. 7.

   Noah says: if m∠ABD + m∠DBC = m∠ABC, then ray BD bisects ∠ABC. Which example shows the claim is false?

   • ∠ABD = 25°, ∠DBC = 25° and ∠ABC = 50°.
   • ∠ABD = 40°, ∠DBC = 40° and ∠ABC = 80°.
   • ∠ABD = 30°, ∠DBC = 50° and ∠ABC = 80°.
   • ∠ABD = 45°, ∠DBC = 45° and ∠ABC = 90°.
9. 8.

   Pia studies the triangle on the card. Which statement is justified?

   • BD bisects AC, but nothing shows it is perpendicular to AC.
   • BD is also perpendicular to AC and bisects AC, because triangles ABD and CBD are congruent by SAS.
   • BD ⊥ AC just when ∠ABC = 90°, since that is the right-angle case.
   • BD ⊥ AC, because an angle bisector is perpendicular to the opposite side of any triangle.
10. 9.

    Ren folds an 8 cm by 6 cm paper rectangle so corner A lands exactly on the opposite corner C. How long is the crease?

    • 10 cm
    • 7.5 cm
    • 6 cm
    • 5 cm
11. 10.

    Towns A and B are 12 km apart on a straight road. Tala places a bus stop equally far from A and B, 8 km off the road. How far is it from A?

    • 14 km
    • 8 km
    • 6 km
    • 10 km