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Geometry and proof · Ages 13–14

The converse of Pythagoras' theorem worksheet

Part of: Pythagoras' theorem

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The converse of Pythagoras' theorem worksheet

Geometry and proof · Ages 13–14

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

Part A: Working out and filling in (2 marks each)

  1. 1.A triangle with sides of 30 cm, 24 cm and 18 cm is right-angled. How long is the side opposite the right angle, in cm?

    Answer:

  2. 2.A triangle's sides are 20 cm, 16 cm and 12 cm. The longest is cm; the two shorter sides' squares add to , and the longest side's square is .

    Fill in each blank

    Answer:

  3. 3.A triangle's sides are 52 cm, 45 cm and 24 cm. The longest is cm; the two shorter sides' squares add to , and the longest side's square is .

    Fill in each blank

    Answer:

  4. 4.In triangle PQR, PQ = 15 cm, QR = 36 cm and PR = 39 cm. Prove ∠PQR = 90°: put the steps in order.

    • A. ∠PQR = 90° (converse of Pythagoras' theorem)
    • B. = 1521 = 392 = PR2
    • C. = 225 + 1296
    • D. PQ2 + QR2 = 152 + 362

    Write the letters in the right order

    Answer:

  5. 5.In triangle PQR, PQ = 16 cm, QR = 30 cm and PR = 34 cm. Prove ∠PQR = 90°: put the steps in order.

    • A. = 256 + 900
    • B. PQ2 + QR2 = 162 + 302
    • C. = 1156 = 342 = PR2
    • D. ∠PQR = 90° (converse of Pythagoras' theorem)

    Write the letters in the right order

    Answer:

Part B: Choose, or true or false (2 marks each)

  1. 6.A triangle with sides of 30 cm, 16 cm and 34 cm is right-angled.

    True or false?

    Answer:

  2. 7.Which of these sets of lengths (in cm) makes a right-angled triangle?

    • A. 41, 42, 58
    • B. 40, 42, 58
    • C. 40, 42, 59
    • D. 40, 43, 59

    Choose the right answer

    Answer:

  3. 8.In triangle ABC, AB = 120 cm, BC = 27 cm and CA = 123 cm. Which angle is the right angle?

    • A. ∠C
    • B. ∠B
    • C. None of them
    • D. ∠A

    Choose the right answer

    Answer:

  4. 9.In triangle ABC, AB = 48 cm, BC = 90 cm and AC = 102 cm. Is ∠ABC a right angle?

    • A. Can't tell without measuring
    • B. No: AB2 + BC2 ≠ AC2
    • C. No: AC isn't the longest side
    • D. Yes: AB2 + BC2 = AC2 (converse of Pythagoras' theorem)

    Choose the right answer

    Answer:

  5. 10.In triangle ABC, AB = 4 cm, BC = 5 cm and CA = 3 cm. Which angle is the right angle?

    • A. ∠B
    • B. None of them
    • C. ∠A
    • D. ∠C

    Choose the right answer

    Answer:

  6. 11.A triangle with sides of 12 cm, 35 cm and 37 cm is right-angled.

    True or false?

    Answer:

  7. 12.A triangle with sides of 84 cm, 77 cm and 36 cm is right-angled.

    True or false?

    Answer:

Part C: Word problems (4 marks each)

  1. 13.A triangular lawn has sides 5 m, 12 m and 13 m. The gardener says: "The angle facing the longest side is a right angle." Is he right?

    • A. No: 52 + 122 ≠ 132
    • B. Yes: the longest side always faces a right angle
    • C. Yes: 52 + 122 = 132, by the converse of Pythagoras' theorem
    • D. Can't tell without measuring

    Choose the right answer

    Working:

    Answer:

  2. 14.A rope 90 m long is split into three pieces in the ratio 5 : 12 : 13 and pulled into a triangle. The pieces are m, m and m, and the triangle's area is m2.

    Fill in each blank

    Working:

    Answer:

  3. 15.A carpenter checks whether a corner of a bookcase is a right angle. Along its two edges he measures 72 cm and 21 cm from the corner, and the two points are 76 cm apart. Is the corner a right angle?

    • A. Yes, as 722 + 212 = 762
    • B. Can't tell without a protractor
    • C. Yes, as 72 + 21 is more than 76
    • D. No, as 722 + 212 ≠ 762

    Choose the right answer

    Working:

    Answer:

  4. 16.A triangular board has sides of 63 cm, 60 cm and 87 cm. What is its area, in cm²?

    Working:

    Answer:

  5. 17.In triangle ABC, BC = 16 cm, AC = 30 cm and AB = 34 cm. Find its area: put the steps in order.

    • A. ∠ACB = 90° (converse of Pythagoras' theorem)
    • B. Area = 16 × 30 ÷ 2 = 240 cm²
    • C. BC2 + AC2 = 256 + 900 = 1156 = AB2

    Write the letters in the right order

    Working:

    Answer:

  6. 18.In the four-sided field ABCD, ∠ABC = 90°, AB = 9 m, BC = 12 m, CD = 20 m and DA = 25 m. AC = m; as AC2 + CD2 = DA2, ∠ACD = 90°; the field's area is m².

    Fill in each blank

    Working:

    Answer:

Show the answers and working

The converse of Pythagoras' theorem worksheet: Answers and working

Part A: Working out and filling in

  1. 1. 30 cm

    The right angle faces the longest side: 242 + 182 = 900 = 302, so it is 30 cm.

  2. 2. 20, 400, 400

    The longest is 20 cm; the other two squared and added make 400, and 202 = 400. Only when the two are equal is the triangle right-angled.

  3. 3. 52, 2601, 2704

    The longest is 52 cm; the other two squared and added make 2601, and 522 = 2704. Only when the two are equal is the triangle right-angled.

  4. 4. D → C → B → A

    Work out the shorter sides' squares added, 1521, which is PR2; the converse of Pythagoras' theorem then gives the right angle, facing PR: ∠PQR.

  5. 5. B → A → C → D

    Work out the shorter sides' squares added, 1156, which is PR2; the converse of Pythagoras' theorem then gives the right angle, facing PR: ∠PQR.

Part B: Choose, or true or false

  1. 6. True

    The longest side is 34 cm: the two shorter sides' squares add to 1156, and 342 = 1156. Only if they are equal is it right-angled (the converse of Pythagoras' theorem).

  2. 7. B. 40, 42, 58

    402 + 422 = 3364 = 582, right-angled by the converse of Pythagoras' theorem; the other three don't match.

  3. 8. B. ∠B

    1202 + 272 = 15129 = 1232, so by the converse of Pythagoras' theorem the right angle faces the longest side, 123 cm.

  4. 9. D. Yes: AB2 + BC2 = AC2 (converse of Pythagoras' theorem)

    AB2 + BC2 = 482 + 902 = 10404 and AC2 = 1022 = 10404. If they are equal, ∠ABC is a right angle by the converse of Pythagoras' theorem; if not, it isn't.

  5. 10. C. ∠A

    42 + 32 = 25 = 52, so by the converse of Pythagoras' theorem the right angle faces the longest side, 5 cm.

  6. 11. True

    The longest side is 37 cm: 122 + 352 = 1369 and 372 = 1369. Only if they are equal is it right-angled.

  7. 12. False

    The longest side is 84 cm: 362 + 772 = 7225 and 842 = 7056. Only if they are equal is it right-angled.

Part C: Word problems

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

  1. 13. C. Yes: 52 + 122 = 132, by the converse of Pythagoras' theorem

    The two shorter sides: 52 + 122 = 169; the longest: 132 = 169. Only if they are equal is it a right angle (converse of Pythagoras' theorem).

  2. 14. 15, 36, 39, 270

    Each share is 90 ÷ 30 = 3 m, so 15, 36 and 39 m. 152 + 362 = 392, so by the converse of Pythagoras' theorem the two shorter sides are perpendicular: area = 15 × 36 ÷ 2 = 270 m2.

  3. 15. D. No, as 722 + 212 ≠ 762

    722 + 212 = 5625 and 762 = 5776: only if equal is it a right angle (converse of Pythagoras' theorem).

  4. 16. 1890 cm²

    632 + 602 = 7569 = 872, so by the converse of Pythagoras' theorem the 63 cm and 60 cm sides are at right angles: area = 63 × 60 ÷ 2 = 1890 cm².

  5. 17. C → A → B

    ∠ACB must be shown to be right first; only then are BC and AC a base and its height: area = 16 × 30 ÷ 2 = 240 cm².

  6. 18. 15, 204

    AC = 92 + 122 = 15 m (Pythagoras' theorem); 152 + 202 = 252, so ∠ACD = 90° (the converse); area = 9 × 12 ÷ 2 + 15 × 20 ÷ 2 = 204 m².

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