Worksheet: The Pythagorean Theorem

Practice

In this worksheet, we will practice using the Pythagorean theorem to find the length of the hypotenuse or a leg of a right triangle and its area.

Q1:

Find 𝑥 in the right triangle shown.

Q2:

Find the area of the triangle 𝐴 𝐵 𝐶 , correct to one decimal place.

Q3:

Given that the triangle 𝐴 𝐵 𝐶 is right-angled at 𝐵 , 𝐴 𝐵 = 1 2 c m , and 𝐵 𝐶 = 5 c m , find the length of 𝐴 𝐶 .

Q4:

Determine the length of the longest side of a right triangle if the lengths of the two other sides are 6 cm and 8 cm.

Q5:

Which of the relations below is correct?

  • A 1 2 1 = 𝑥 + 3 6
  • B 𝑥 = 6 + 1 1
  • C 𝑥 = ( 1 1 + 6 )
  • D 𝑥 = 6 + 1 1

Q6:

Given that 𝐴 𝐵 𝐶 is a right-angled triangle, where 𝐶 𝐵 = 8 c m and 𝐴 𝐶 = 1 0 c m , determine the length of 𝐴 𝐵 .

Q7:

Find 𝑥 in the right triangle shown.

Q8:

Find 𝑥 in the right triangle shown.

Q9:

𝐴 𝐵 𝐶 is a right-angled triangle at 𝐵 , where 𝐴 𝐵 = 1 2 6 c m and 𝐵 𝐶 = 7 8 . 4 c m . Find the length of 𝐴 𝐶 .

Q10:

Determine the diagonal length of the rectangle whose length is 48 cm, and width is 20 cm.

Q11:

Determine, to the nearest tenth, the perimeter of a right triangle, given that its hypotenuse is 49 cm and that one of its sides is 26 cm.

Q12:

Given that 𝐴 𝐵 𝐶 𝐷 is a square, find the length of 𝐷 𝐸 .

Q13:

Determine the area of 𝐴 𝐵 𝐶 𝐷 .

Q14:

Find 𝑥 in the right triangle shown.

Q15:

Find 𝑥 in the right triangle shown.

Q16:

Find 𝑥 in the right triangle shown.

Q17:

Find 𝑥 in the right triangle shown.

Q18:

Find 𝑥 in the right triangle shown.

Q19:

Find 𝑥 in the right triangle shown.

Q20:

Find 𝑥 in the right triangle shown.

Q21:

In a right-angled triangle, which of the following is equal to the area of the square that has the hypotenuse as a side?

  • AThe area of the square drawn on one of the other two sides.
  • BThe sum of the perimeters of the squares drawn on the other two sides.
  • CThe sum of the areas of the squares drawn on the other two sides.

Q22:

Find 𝑥 in the right triangle shown.

Q23:

A triangle has vertices at the points 𝐴 ( 4 , 1 ) , 𝐵 ( 6 , 2 ) , and 𝐶 ( 2 , 5 ) .

Work out the lengths of the sides of the triangle. Give your answers as surds in their simplest form.

  • A 𝐴 𝐵 = 5 , 𝐴 𝐶 = 4 √ 2 , and 𝐵 𝐶 = √ 7
  • B 𝐴 𝐵 = 5 , 𝐴 𝐶 = 5 √ 2 , and 𝐵 𝐶 = 5
  • C 𝐴 𝐵 = √ 5 , 𝐴 𝐶 = 2 √ 3 , and 𝐵 𝐶 = √ 7
  • D 𝐴 𝐵 = √ 5 , 𝐴 𝐶 = 2 √ 5 , and 𝐵 𝐶 = 5
  • E 𝐴 𝐵 = 5 , 𝐴 𝐶 = 5 √ 2 , and 𝐵 𝐶 = √ 5

Is the triangle a right triangle?

  • Ayes
  • BIt is impossible to tell.
  • Cno

Q24:

Find 𝑥 in the right triangle shown.

Q25:

Find the length of the hypotenuse of a right triangle whose leg lengths are 1.

  • A1
  • B2
  • C √ 3
  • D √ 2
  • E3

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