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?

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

3. Use this

   Proof card. Given: A, B and F lie on a line. Ray BD bisects angle ABC. Ray BE bisects angle CBF. Prove: m∠DBE = 90°. Step 1: m∠DBC = ½ m∠ABC and m∠CBE = ½ m∠CBF. Step 2: m∠DBE = m∠DBC + m∠CBE. Step 3: m∠DBE = ½(m∠ABC + m∠CBF). Step 4: m∠DBE = ½(180°) = 90°.

4. 2.

   Bo writes the proof shown on the card. Which fact makes Step 4 work?

   • ∠ABC and ∠CBF form a straight angle, so they add to 180°.
   • Two bisectors of adjacent angles are perpendicular to each other, whatever the angles are.
   • Adjacent angles are complementary, so they add to 90° and the two halves add to 45°.
   • BD and BE are both bisectors, so ∠DBC = ∠CBE.
5. 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?

   • 4.5 cm
   • 6 cm
   • 24 cm
   • 18 cm
6. 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?

   • 29°
   • 116°
   • 8°
   • 58°
7. 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°
   • 33°
   • 22°
   • 44°
8. 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?

   • 13
   • 11
   • 9
   • 22
9. 7.

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

   • ∠ABD = 30°, ∠DBC = 50° and ∠ABC = 80°.
   • ∠ABD = 40°, ∠DBC = 40° and ∠ABC = 80°.
   • ∠ABD = 25°, ∠DBC = 25° and ∠ABC = 50°.
   • ∠ABD = 45°, ∠DBC = 45° and ∠ABC = 90°.
10. 8.

    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
    • 6 cm
    • 7.5 cm
    • 5 cm
11. 9.

    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
12. 10.

    Zee draws a chord AB, 16 cm long, in a circle of radius 10 cm. How far is the chord from the center of the circle?

    • 8 cm
    • 18 cm
    • 10 cm
    • 6 cm