Download 8-2 Special Right Triangles

8-2 Special Right
Triangles
45˚-45˚-90˚and 30˚-60˚-90˚
45˚-45˚-90˚ Triangle
• Two angles of a particular right triangle are 45˚
• What else can we say about this triangle, besides that it is a
right triangle?
• If each leg of the triangle has a measure of x length, what is
the measure of the hypotenuse?
45˚-45˚-90˚ Triangle Theorem
• We have just proved the 45˚-45˚-90˚ Triangle Theorem:
• In a 45˚-45˚-90˚ triangle, both legs are congruent and the
length of the hypotenuse is 2 times longer than the length of
a leg.
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 2 ∙ 𝑙𝑒𝑔
Find the value of each variable
• A.
• B.
More Practice:
• Find the length of the hypotenuse of a 45˚-45˚-90˚ triangle
with legs of length 5 3.
• Find the length of a leg of a 45˚-45˚-90˚ triangle with a
hypotenuse of length 10.
• A square garden has sides 100ft long. You want to build a brick
path along a diagonal of the square. To the nearest foot, how
long will the path be?
30˚-60˚-90˚ Triangle Theorem
• A similar relationship exists within right
triangles with angle measures of 30˚ and 60˚.
• The 30˚-60˚-90˚ Triangle Theorem says:
• In a 30˚-60˚-90˚ triangle, the length of the
hypotenuse is twice the length of the shorter
leg. The length of the longer leg is 3 times
longer than the length of the shorter leg.
ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒 = 2 ∙ 𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔
𝑙𝑜𝑛𝑔𝑒𝑟 𝑙𝑒𝑔 = 3 ∙ 𝑠ℎ𝑜𝑟𝑡𝑒𝑟 𝑙𝑒𝑔
30˚-60˚-90˚ Triangle Theorem: Proof
• ∆𝑊𝑋𝑌 is a 30˚-60˚-90˚ triangle within equilateral ∆𝑊𝑋𝑍.
• If 𝑋𝑌 = 𝑙𝑒𝑛𝑔𝑡ℎ 𝑠, what is the length of 𝑊𝑋?
• Use the Pythagorean theorem to solve for the length of
𝑊𝑋.
Using 30˚-60˚-90˚ Triangle Theorem
• Find the value of each variable.
More Practice
• In quadrilateral ABCD, AD = DC and AC = 20. Find the area of
ABCD. Leave your answer in simplest radical form.