Geometry and proof · Ages 14–15

The mid-point theorem worksheet

Part of: Quadrilaterals

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Geometry and proof · Ages 14–15

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.In the picture, M is the mid-point of AB and N the mid-point of AC. x =
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2.In the picture, M is the mid-point of AB and N the mid-point of AC. x =
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3.In the picture, M is the mid-point of AB and N the mid-point of AC. x =
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4.In △ABC, M and N are the mid-points of AB and AC, MN = (x + 4) cm and BC = (4x − 2) cm. x =
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5.In the picture, M is the mid-point of AB, MN // BC, AC = 10 cm and BC = 8 cm. AN = cm, MN = cm
Fill in each blank
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6.In △ABC, M and N are the mid-points of AB and AC, MN = (x + 4) cm and BC = (4x − 4) cm. x =
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7.In the picture, M is the mid-point of AB, MN // BC, AC = 10 cm and BC = 10 cm. AN = cm, MN = cm
Fill in each blank
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8.In △ABC, M and N are the mid-points of AB and AC, MN = (x + 1) cm and BC = (3x − 7) cm. x =
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9.In the trapezium ABCD, AB // DC, and M and N are the mid-points of AD and BC. x =
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10.The quadrilateral ABCD has diagonals AC = 17 cm and BD = 16 cm. P, Q, R and S are the mid-points of AB, BC, CD and DA. The perimeter of PQRS =
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11.The quadrilateral ABCD has diagonals AC = 16 cm and BD = 12 cm. P, Q, R and S are the mid-points of AB, BC, CD and DA. The perimeter of PQRS =
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12.The quadrilateral ABCD has diagonals AC = 17 cm and BD = 12 cm. P, Q, R and S are the mid-points of AB, BC, CD and DA. The perimeter of PQRS =
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13.A triangular roof truss ABC has a base BC of 60 dm, its top A 48 dm above the base. A tie beam joins the mid-points M and N of the two rafters. The beam MN = dm, and it is dm above BC.
Fill in each blank
Working:
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14.A triangular piece of cloth has sides of 16 cm, 20 cm and 12 cm and an area of 8 cm². The middle triangle joining its sides' mid-points is cut out. Its perimeter = cm and its area = cm².
Fill in each blank
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15.Points B and C on either side of a pond can't be measured directly. From a point A on the bank, the mid-points M and N of AB and AC are 34 m apart. How far apart are B and C?
Working:
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16.A stepladder's frame is a trapezium 46 cm wide at the bottom and 26 cm at the top; a bar joins the mid-points of the two sides, parallel to the bottom. How long is the bar?
Working:
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17.A triangular piece of cloth has sides of 16 cm, 16 cm and 16 cm and an area of 8 cm². The middle triangle joining its sides' mid-points is cut out. Its perimeter = cm and its area = cm².
Fill in each blank
Working:
Answer:
18.A triangular roof truss ABC has a base BC of 110 dm, its top A 33 dm above the base. A tie beam joins the mid-points M and N of the two rafters. The beam MN = dm, and it is dm above BC.
Fill in each blank
Working:
Answer:

Show the answers and working

The mid-point theorem worksheet: Answers and working

1. 4 cm
Mid-point theorem: MN = BC ÷ 2, so x = 4.
2. 6 cm
Mid-point theorem: BC = 2 × MN = 2 × 3 = 6.
3. 55°
MN // BC (mid-pt. thm.), so x = ∠ABC = 55° (corr. ∠s, MN // BC).
4. 5
BC = 2 × MN (mid-pt. thm.): 4x − 2 = 2(x + 4), x = 5. Check: MN = 9 cm.
5. 5, 4
M is the mid-point and MN // BC, so N is the mid-point of AC (intercept thm.): AN = 10 ÷ 2 = 5; MN = 8 ÷ 2 = 4 (mid-pt. thm.).
6. 6
BC = 2 × MN (mid-pt. thm.): 4x − 4 = 2(x + 4), x = 6. Check: MN = 10 cm.
7. 5, 5
M is the mid-point and MN // BC, so N is the mid-point of AC (intercept thm.): AN = 10 ÷ 2 = 5; MN = 10 ÷ 2 = 5 (mid-pt. thm.).
8. 9
BC = 2 × MN (mid-pt. thm.): 3x − 7 = 2(x + 1), x = 9. Check: MN = 10 cm.
9. 9 cm
Join AC, meeting MN at P: MP = 5 ÷ 2 (mid-pt. thm. in △ACD) and PN = 13 ÷ 2 (in △ABC), so x = (13 + 5) ÷ 2 = 9.
10. 33 cm
In △ABC, PQ = AC ÷ 2, and in △ACD, SR = AC ÷ 2; likewise QR = PS = BD ÷ 2 (mid-pt. thm.): perimeter = 17 + 16 = 33 cm.
11. 28 cm
In △ABC, PQ = AC ÷ 2, and in △ACD, SR = AC ÷ 2; likewise QR = PS = BD ÷ 2 (mid-pt. thm.): perimeter = 16 + 12 = 28 cm.
12. 29 cm
In △ABC, PQ = AC ÷ 2, and in △ACD, SR = AC ÷ 2; likewise QR = PS = BD ÷ 2 (mid-pt. thm.): perimeter = 17 + 12 = 29 cm.

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 30, 24
Mid-point theorem: MN // BC and MN = 60 ÷ 2 = 30 dm; M is halfway along AB, so it is halfway up too: 48 ÷ 2 dm.
14. 24, 2
Each side is half the side facing it (mid-pt. thm.): perimeter = (16 + 20 + 12) ÷ 2 = 24 cm; the four small triangles are congruent, so area = 8 ÷ 4 = 2 cm².
15. 68 m
Mid-point theorem: BC = 2 × MN = 2 × 34 = 68 m.
16. 36 cm
Draw a diagonal and use the mid-point theorem twice: 46 ÷ 2 + 26 ÷ 2 = (46 + 26) ÷ 2 = 36 cm.
17. 24, 2
Each side is half the side facing it (mid-pt. thm.): perimeter = (16 + 16 + 16) ÷ 2 = 24 cm; the four small triangles are congruent, so area = 8 ÷ 4 = 2 cm².
18. 55, 16.5
Mid-point theorem: MN // BC and MN = 110 ÷ 2 = 55 dm; M is halfway along AB, so it is halfway up too: 33 ÷ 2 dm.

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