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.


Find the value of 𝑥 in the given figure.


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


Find the length of 𝐴𝐶.


Find the values of 𝑎 and 𝑏.

  • A𝑎=36, 𝑏=3
  • B𝑎=6, 𝑏=36
  • C𝑎=6, 𝑏=63
  • D𝑎=63, 𝑏=6
  • E𝑎=3, 𝑏=36


Find the length of 𝐴𝐵.


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


In triangle 𝐴𝐵𝐶, the length of 𝐵𝐶 is 𝑥.

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

  • A2𝑥
  • B𝑥2
  • C𝑥2
  • D2𝑥
  • E𝑥

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

  • A3𝑥
  • B3𝑥
  • C32𝑥
  • D𝑥3
  • E𝑥3


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


Find the lengths of 𝐵𝐷 and 𝐷𝐸.

  • A𝐵𝐷=73.95cm, 𝐷𝐸=37cm
  • B𝐵𝐷=32.87cm, 𝐷𝐸=10.96cm
  • C𝐵𝐷=49.3cm, 𝐷𝐸=24.65cm
  • D𝐵𝐷=59.16cm, 𝐷𝐸=19.72cm


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


Find the length of 𝑌𝑍.


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


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


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.


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


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


Triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹 are similar. What is the length of 𝐸𝐹?


Find the value of 𝑥.

  • A194
  • B2
  • C254
  • D72

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