- Identify and use the ratios involved with right isosceles triangles.
Right Isosceles Triangles
- Each triangle has the angles 45∘,45∘ (from the two angles cut in half), and ___________.
The diagonal becomes the hypotenuse of each isosceles right triangle because it is across from the right angle. Since a square has 4 congruent sides, each triangle is isosceles where the legs are the congruent sides of the square.
- The diagonal of the square becomes the __________________________ of each triangle.
In the diagram above, the ______________ and the base ________________ are each congruent.
Don’t forget that the base angles are the angles that are opposite the congruent sides. They don’t have to be on the bottom of the figure, like in the picture below:
The isosceles right triangle below has legs measuring 1 centimeter.
Use the Pythagorean Theorem to find the length of the hypotenuse.
What if each leg in the example above was 5 cm? Then we would have:
What does proportional mean?
You may recognize the word “proportion,” which means “ratio” or “fraction.”
“Proportional” describes a relationship between 2 values where you can multiply one of the values by some number and get the second value.
For instance, 3 and 6 have the same “proportional” relationship as 4 and 8, because you need to multiply the first number by 2 to get the second number in both cases.
Another pair of numbers with the same proportional relationship is ________ and _________.
Another example is the sentence: “Punishment should be proportional to the crime”.
This means that the worse a crime is, the harsher the punishment should be.
As we discovered in the examples on the previous page,
- The length of the ___________________________ equals 2√ times the length of a leg.
This relationship is very important to know!
What is the length of the hypotenuse in the triangle below?
First, we must determine which side of the triangle is the hypotenuse.
This makes the legs the other two sides, which have a length of _________________.
1. Every isosceles right triangle has 3 special interior angles. What are they?
__________ , __________ , and __________
2. If an isosceles right triangle has legs that are 3 inches long, how long is its hypotenuse?
a. Draw a picture of the triangle here:
b. Use the Pythagorean Theorem to find the length of the hypotenuse (like in Example 1):
c. Use the special proportional relationship to find the length of the hypotenuse (like in Example 2):
d. Are your answers to (b.) and (c.) above the same?
Antonio built a square patio in his backyard.
He wants to make a water pipe for flowers that goes from one corner to another, diagonally. How long will that pipe be?
The first step in a word problem like this is to add important information to the drawing. Because the problem asks you to find the length from one corner to another, you should draw a diagonal line segment (from one corner of the square to the opposite corner) into your patio picture:
Once you draw the diagonal path, you can see how triangles help answer this question.
You cook a grilled cheese sandwich. To make it easier to eat, you cut the sandwich in half diagonally. If each slice of bread (before it is cut) measures 14 cm by 14 cm, how long is the diagonal of your sandwich?
(Hint: draw yourself a picture to start this problem! If you are stuck, look at Example 3 to help you.)