Download Special Right Triangles Worksheet

Mrs. Kukreja
Pre-calculus
SPECIAL RIGHT TRIANGLES
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45º-45º-90º Triangles
A 45- 45- 90 triangle is a special right triangle whose angles are 45, 45and 90. The lengths
of the sides of a 45- 45- 90 triangle are in the ratio of
.
A right triangle with two sides of equal lengths is a 45- 45- 90 triangle.
Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two
sides are both 3 inches.
Solution:
Step 1: This is a right triangle with two equal sides so it must be a 45°- 45°- 90° triangle.
Step 2: You are given that the both the sides are 3. If the first and second value of the ratio
is 3 then the length of the third side is
Answer: The length of the hypotenuse is
inches.
You can also recognize a 45°- 45°- 90° triangle by the angles. As long as you know that one of
the angles in the right-angle triangle is 45° then it must be a 45°- 45°- 90° special right triangle.
A right triangle with a 45° angle must be a 45°- 45°- 90° special right triangle.
Example 2: Find the lengths of the other two sides of a right triangle if the length of the
hypotenuse is
inches and one of the angles is 45°.
Solution:
Step 1: This is a right triangle with a 45°so it must be a 45°- 45°- 90° triangle.
You are given that the hypotenuse is
. If the third value of the ratio
the lengths of the other two sides must 4.
is
then
Answer: The lengths of the two sides are both 4 inches.
30º-60º-90º Triangles
Another type of special right triangles is the 30- 60- 90 triangle. This is right triangle whose
angles are 30, 60and 90. The lengths of the sides of a 30- 60- 90 triangle are in the ratio of
.
Example 1: Find the length of the hypotenuse of a right triangle if the lengths of the other two
sides are 4 inches and
inches.
Solution:
Step 1: Test the ratio of the lengths to see if it fits the
Step 2: Yes, it is a 30°- 60°- 90° triangle for n = 4
Step 3: Calculate the third side.
ratio.
2n = 2×4 = 8
Answer: The length of the hypotenuse is 8 inches.
You can also recognize a 30°- 60°- 90° triangle by the angles. As long as you know that one of
the angles in the right-angle triangle is either 30° or 60° then it must be a 30°- 60°- 90° special
right triangle.
A right triangle with a 30° angle or 60° angle must be a 30°- 60°- 90° special right triangle.
Example 2: Find the lengths of the other two sides of a right triangle if the length of the
hypotenuse is 8 inches and one of the angles is 30°.
Solution:
Step 1: This is a right triangle with a 30° angle so it must be a 30°- 60°- 90° triangle.
You are given that the hypotenuse is 8. Substituting 8 into the third value of the ratio
, we get that 2n = 8  n = 4.
Substituting n = 4 into the first and second value of the ratio we get that the other two sides are 4
and
.
Answer: The lengths of the two sides are 4 inches and
inches.
Special Right Triangles Worksheet
Exercises 1-6 refer to the 30-60-90 triangle. Using the given information, find the indicated
length.
1. AB=14; BC=
2. BC=7; AB=
3. AC=
; BC=
4. AC=
; AB=
Exercises 7-12 refer to the 45-45-90 triangle. Using the given information, find the indicated
length.
7. XY=7; XZ=
9. XZ=
; YZ=
11. YZ=
; XZ=
13. The length of the hypotenuse of a 30-60-90 triangle is 20. What is the length of the shorter
leg?
14. A ladder leaning against a wall makes a 60 angle with the ground. The base of the ladder is 3
m from the building. How high above the ground is the top of the ladder?