Worksheet: Special Right Triangles

In this worksheet, we will practice finding lengths in 30-60-90-degree and in 45-45-90-degree triangles without trigonometry.

Q1:

Find the value of 𝑥 in the given figure.

Q2:

Find the length of 𝐵𝐶 in the figure to two decimal places.

Q3:

Find the length of 𝐴𝐶.

Q4:

Find the values of 𝑎 and 𝑏.

  • A 𝑎 = 3 6 , 𝑏 = 3
  • B 𝑎 = 6 , 𝑏 = 3 6
  • C 𝑎 = 6 , 𝑏 = 6 3
  • D 𝑎 = 6 3 , 𝑏 = 6
  • E 𝑎 = 3 , 𝑏 = 3 6

Q5:

Find the length of 𝐴𝐵.

Q6:

Consider triangle 𝐴𝐵𝐶.

Find, in the form 1 : 𝑥, the ratio of the side lengths 𝐵𝐶 : 𝐴𝐵.

  • A1 : 2
  • B1 : 3
  • C1 :  3
  • D1 :  1
  • E1 : 2

Find, in the form 1 : 𝑥, the ratio of the side lengths 𝐵𝐶 : 𝐴𝐶.

  • A1 : 1
  • B1 : 2
  • C1 : 3
  • D1 : 3
  • E1 : 2

Q7:

In triangle 𝐴𝐵𝐶, the length of 𝐵𝐶 is 𝑥.

Find, in terms of 𝑥, the length of 𝐴𝐶.

  • A 2 𝑥
  • B 𝑥 2
  • C 𝑥 2
  • D 2 𝑥
  • E 𝑥

Find, in terms of 𝑥, the length of 𝐴𝐵.

  • A 3 𝑥
  • B 3 𝑥
  • C 3 2 𝑥
  • D 𝑥 3
  • E 𝑥 3

Q8:

Given that 𝐴𝐸=42cm, and 𝐵𝐷 is a median in 𝐴𝐵𝐶, find the perimeter of 𝐴𝐵𝐷.

Q9:

Find the lengths of 𝐵𝐷 and 𝐷𝐸.

  • A 𝐵 𝐷 = 7 3 . 9 5 c m , 𝐷 𝐸 = 3 7 c m
  • B 𝐵 𝐷 = 3 2 . 8 7 c m , 𝐷 𝐸 = 1 0 . 9 6 c m
  • C 𝐵 𝐷 = 4 9 . 3 c m , 𝐷 𝐸 = 2 4 . 6 5 c m
  • D 𝐵 𝐷 = 5 9 . 1 6 c m , 𝐷 𝐸 = 1 9 . 7 2 c m

Q10:

Given a right-angled triangle 𝐴𝐵𝐶 in which angle 𝐴 measures 30, which of the following is equal to the length of the side opposite to angle 𝐴?

  • Athe length of the hypotenuse
  • Bone-third the length of the hypotenuse
  • Chalf the length of the side opposite to the angle measuring 60
  • Dhalf the length of the hypotenuse

Q11:

Find the length of 𝑌𝑍.

Q12:

In the figure below, 𝑚𝑋𝑌𝑍=𝑚𝑀𝑌𝑁=90, 𝑀 is the midpoint of 𝑋𝑍, and 𝑚𝑁=30. Given that 𝑋𝑍=13cm, find the length of 𝑀𝑁.

  • A14 cm
  • B12 cm
  • C13 cm
  • D11.26 cm
  • E16.25 cm

Q13:

Given that 𝐴𝐵𝐶𝐷 is a rectangle, find the length of 𝐶𝐸.

Q14:

If the perimeter of a rhombus is 88 cm, and the measure of one of its angles is 60, find its area approximated to the nearest integer.

  • A484 cm2
  • B838 cm2
  • C242 cm2
  • D419 cm2

Q15:

In the trapezoid 𝐴𝐵𝐶𝐷, 𝑚𝐵𝐴𝐻=30. Find the area of 𝐴𝐵𝐶𝐷 in square feet. Round your answer to the nearest tenth.

Q16:

Use the information in the figure to determine the area of the square.

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