Transcript Warm Up Find x. Leave answer as simplified radical ANSWER: 27 5
Warm Up
Find x. Leave answer as simplified radical ANSWER: 27 5
Special Right Triangles
30 – 60 – 90
30º
Find the missing side lengths
10 60º • Can we use Pythagorean Theorem?
• Can we use similar Triangles?
• Any other way we know of to find the missing side lengths?
Before we begin…
Lets go over some vocabulary needed ALTITUDE
The perpendicular height from one side of a triangle to the opposite vertex
HYPOTENUSE
The longest side of a right triangle (the side across from the right angle)
LEG
The two sides that connect to the right angle in a right triangle.
Discovering Special Triangles
1. Adam, a construction manager in a nearby town, needs to check the uniformity of Yield signs around the state and is checking the heights (altitudes) of the Yield signs in your locale. Adam knows that all yield signs have the shape of an equilateral triangle. Why is it sufficient for him to check just the heights (altitudes) of the signs to verify uniformity?
Because all equilateral triangles are similar so one measurement will be sufficient.
Discovering Special Triangles
2. A Yield sign from a street near your home is pictured to the right. It has the shape of an equilateral triangle with a side length of 2 feet. If the altitude of the triangular sign is drawn, you split the Yield sign in half vertically, creating two 30°-60° 90° right triangles, as shown to the right. For now, we’ll focus on the right triangle on the right side. (We could just as easily focus on the right triangle on the left; we just need to pick one.) We know that the hypotenuse is 2 ft., that information is given to us. The shorter leg has length 1 ft. Why?
2 2 Congruent due to HL 2 2 So the two bottom legs must be congruent 2 1 1 2 We know all sides have a length of 2. So if that side is split into 2 congruent pieces each piece must be 1.
3.What is the length of the third side (the altitude)? Leave answer as simplified radical.
Pythagorean Theorem: 1 2 + x 2 = 2 2 1 + x 2 = 4 x 2 = 3 x =
3 2 1 X 1 2
Quick Review: How do we simplify radicals?
Break down the radicand (the number inside the radical) into perfect squares. Anything that is a perfect square will come out of the radical everything else stays inside the radical.
120 12 6 2 5 10 2 3 2 so 120 2 30
Quick Review: How do you rationalize the denominator?
We can never leave a radical in the denominator. Multiply the numerator and denominator by the radical on the bottom. This will get rid of the radical on the denominator, then simplify. 18 3 ∙ 3 3 = 18 3 = 6 3 3
Answer question 4 on your own or in your pair
5) Now that we have found the altitudes of both equilateral triangles, we look for patterns in the data. Fill in the first two rows of the chart below, and write down any observations you make. Then fill in the third and fourth rows. Each 30°- 60°- 90° right triangle formed by drawing altitude Hypotenuse Side Length of Equilateral Triangle Length 2 (first) 1 (second) 4 6 Shorter Leg Length Longer Leg Length
5) Now that we have found the altitudes of both equilateral triangles, we look for patterns in the data. Fill in the first two rows of the chart below, and write down any observations you make. Then fill in the third and fourth rows. Each 30°- 60°- 90° right triangle formed by drawing altitude Hypotenuse Side Length of Equilateral Triangle Length 2 (first)
2
1 (second)
1
4 6
4 6
Shorter Leg Length Longer Leg Length
1 1/2 2 3
2 1 2 3 3 3 3 3
6. What is true about the lengths of the sides of any 30°-60°-90° right triangle? 30º 2x x 3 x 60º