8.4: Special Right Triangles
Learning Objectives
 Identify and use the ratios involved with isosceles right triangles.
 Identify and use the ratios involved with 306090 triangles.
Review Queue
Find the value of the missing variable(s). Simplify all radicals.
 Do the lengths 6, 6, and
62√ make a right triangle?  Do the lengths
3,33√, and 6 make a right triangle?
Know What? The Great Giza Pyramid is a pyramid with a square base and four isosceles triangles that meet at a point. It is thought that the original height was 146.5 meters and the base edges were 230 meters.
First, find the length of the edge of the isosceles triangles. Then, determine if the isosceles triangles are also equilateral triangles. Round your answers to the nearest tenth.
You can assume that the height of the pyramid is from the center of the square base and is a vertical line.
Isosceles Right Triangles
There are two types of special right triangles, based on their angle measures. The first is an isosceles right triangle. Here, the legs are congruent and, by the Base Angles Theorem, the base angles will also be congruent. Therefore, the angle measures will be
Investigation 82: Properties of an Isosceles Right Triangle
Tools Needed: Pencil, paper, compass, ruler, protractor
 Construct an isosceles right triangle with 2 in legs. Use the SAS construction that you learned in Chapter 4.
 Find the measure of the hypotenuse. What is it? Simplify the radical.
 Now, let’s say the legs are of length
x and the hypotenuse ish . Use the Pythagorean Theorem to find the hypotenuse. What is it? How is this similar to your answer in #2?
454590 Corollary: If a triangle is an isosceles right triangle, then its sides are in the extended ratio
Step 3 in the above investigation proves the 454590 Triangle Theorem. So, anytime you have a right triangle with congruent legs or congruent angles, then the sides will always be in the ratio
Example 1: Find the length of the missing sides.
a)
b)
Solution: Use the
a)
b)
Example 2: Find the length of
a)
b)
Solution: Again, use the
a)
b)
In part b, we rationalized the denominator. Whenever there is a radical in the denominator of a fraction, multiply the top and bottom by that radical. This will cancel out the radical from the denominator and reduce the fraction.
306090 Triangles
The second special right triangle is called a 306090 triangle, after the three angles. To construct a 306090 triangle, start with an equilateral triangle.
Investigation 83: Properties of a 306090 Triangle
Tools Needed: Pencil, paper, ruler, compass
1. Construct an equilateral triangle with 2 in sides.
2. Draw or construct the altitude from the top vertex to the base for two congruent triangles.
3. Find the measure of the two angles at the top vertex and the length of the shorter leg.
The top angles are each
4. Find the length of the longer leg, using the Pythagorean Theorem. Simplify the radical.
5. Now, let’s say the shorter leg is length
306090 Corollary: If a triangle is a 306090 triangle, then its sides are in the extended ratio
Step 5 in the above investigation proves the 306090 Corollary. The shortest leg is always
Example 3: Find the length of the missing sides.
a)
b)
Solution: In part a, we are given the shortest leg and in part b, we are given the hypotenuse.
a) If
b) Now,
Example 4: Find the value of
a)
b)
Solution: In part a, we are given the longer leg and in part b, we are given the hypotenuse.
a)
b)
Example 5: Find the measure of
Solution: Think of this trapezoid as a rectangle, between a 454590 triangle and a 306090 triangle.
From this picture,
Know What? Revisited The line that the vertical height is perpendicular to is the diagonal of the square base. This length (blue) is the same as the hypotenuse of an isosceles right triangle because half of a square is an isosceles right triangle. So, the diagonal is
\begin{align*}edge = \sqrt{\left (115 \sqrt{2} \right )^2 + 146.5^2} \approx 218.9 \ m\end{align*}
In order for the sides to be equilateral triangles, this length should be 230 meters. It is not, so the triangles are isosceles.
Review Questions
 In an isosceles right triangle, if a leg is \begin{align*}x\end{align*}
x , then the hypotenuse is __________.  In a 306090 triangle, if the shorter leg is \begin{align*}x\end{align*}
x , then the longer leg is __________ and the hypotenuse is ___________.  A square has sides of length 15. What is the length of the diagonal?
 A square’s diagonal is 22. What is the length of each side?
 A rectangle has sides of length 4 and \begin{align*}4 \sqrt{3}\end{align*}
43√ . What is the length of the diagonal?  A baseball diamond is a square with 90 foot sides. What is the distance from home base to second base? (HINT: It’s the length of the diagonal).
For questions 718, find the lengths of the missing sides.
 Do the lengths \begin{align*}8 \sqrt{2}, 8 \sqrt{6}\end{align*}
82√,86√ , and \begin{align*}16 \sqrt{2}\end{align*}162√ make a special right triangle? If so, which one?  Do the lengths \begin{align*}4 \sqrt{3}, 4 \sqrt{6}\end{align*}
43√,46√ and \begin{align*}8 \sqrt{3}\end{align*}83√ make a special right triangle? If so, which one?  Find the measure of \begin{align*}x\end{align*}
x .  Find the measure of \begin{align*}y\end{align*}
y .  What is the ratio of the sides of a rectangle if the diagonal divides the rectangle into two 306090 triangles?
 What is the length of the sides of a square with diagonal 8 in?
For questions 2528, it might be helpful to recall #25 from section 8.1.
 What is the height of an equilateral triangle with sides of length 3 in?
 What is the area of an equilateral triangle with sides of length 5 ft?
 A regular hexagon has sides of length 3 in. What is the area of the hexagon? (Hint: the hexagon is made up a 6 equilateral triangles.)
 The area of an equilateral triangle is \begin{align*}36 \sqrt{3}\end{align*}
363√ . What is the length of a side?  If a road has a grade of \begin{align*}30^\circ\end{align*}
30∘ , this means that its angle of elevation is \begin{align*}30^\circ\end{align*}30∘ . If you travel 1.5 miles on this road, how much elevation have you gained in feet (5280 ft = 1 mile)?  Four isosceles triangles are formed when both diagonals are drawn in a square. If the length of each side in the square is \begin{align*}s\end{align*}
s , what are the lengths of the legs of the isosceles triangles?
Review Queue Answers

\begin{align*}4^2 + 4^2 = x^2\!\\
{\;} \quad \ \ 32 = x^2\!\\
{\;} \qquad x = 4 \sqrt{2}\end{align*}
42+42=x2 32=x2x=42√ 
\begin{align*}3^2 + z^2 = 6^2 \qquad \left( 3 \sqrt{3} \right)^2 + 9^2 = y^2\!\\
{\;} \qquad z^2 = 27 \qquad \qquad \quad \ \ 108=y^2\!\\
{\;} \qquad \ z = 3 \sqrt{3} \qquad \qquad \quad \ \ y = 6 \sqrt{3}\end{align*}
32+z2=62(33√)2+92=y2z2=27 108=y2 z=33√ y=63√ 
\begin{align*}x^2 + x^2 = 10^2\!\\
{\;} \ \ \quad 2x^2 = 100\!\\
{\;} \qquad x^2 = 50\!\\
{\;} \qquad \ x = 5 \sqrt{2}\end{align*}
x2+x2=102 2x2=100x2=50 x=52√  Yes, \begin{align*}6^2 + 6^2 = \left(6 \sqrt{2} \right)^2 \rightarrow 36 + 36 = 72\end{align*}
62+62=(62√)2→36+36=72  Yes, \begin{align*}3^2 + \left(3 \sqrt{3} \right)^2 = 6^2 \rightarrow 9 + 27 = 36\end{align*}
32+(33√)2=62→9+27=36
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