Real World Applications

Today we learned about triangles in the real world.

Objective:
Students will solve real world problems using triangles.

TEKS:
G.4

The student is expected to select an appropriate
representation (concrete, pictorial, graphical, verbal, or
symbolic) in order to solve problems.

Mr. Escalante described lots of ways to use triangles to solve problems. He said that construction workers, painters, engineers, gardeners and architects all apply the things we have learned about triangles on a regular basis.

For example, painters need to be able to find the areas of things that they will paint, and many houses have triangular portions below the roof.

                         

Engineers use perimeters find out how much material is needed to build a truss.

Construction workers use special right triangles, the Pythagorean theorem and similarity to construct pitched roofs.

As you can see, there are lots of applications of all the things we learned about triangles in this unit.

Special Right triangles

Today the lesson was on special right triangles.

Today’s Objective: Students will be able to label each side of the special right triangle given the angle measurements and one side length.

Today’s TEKS: G.5(D) identify and apply patterns from right triangles to solve meaningful problems, including special right triangles (45-45-90 and 30-60-90).

There are two types of special right triangles that we learned about.

45-45-90 Triangle – Given an isosceles right triangle, the two legs are equal to each other (say x) and the hypotenuse is equal to the square root of 2 multiplied by the leg length (x√2).

30-60-90 Triangle – Given a right triangle with angle measures 30°, 60°, and 90° the leg across from the 30° angle has the shortest side length (say x), the leg across from the 60° angle has a side length equal to the square root of three multiplied by the shortest leg length (x√3), and the hypotenuse is double the length of the shortest leg length (2x).

Question:  If I have a 45-45-90 triangle and I am given the hypotenuse has length 3, how can I find the length of each leg?

Area and Perimeter

Today's objective was: The student will be able to evaluate area and perimeter using given information.

Todays TEKS: G.8.A  The student is expected to: find areas of regular polygons, circles, and
composite figures

Today we learned about the area and perimeter of triangles.  The area of a triangle is A=(BH)/2 and to get the perimeter you add all of the side together.  Most of us had already learned this so we did an activity where we split into groups and came up with real world uses for area and perimeter of triangles.  We then all presented our examples to the class so we could see what other groups came up with and different ways area and perimeter can be used.  After we discussed these examples, Mr. Escalantes gave us some problems where we didn't have all of the sides and we had to use the Pythagorean Theorem we learned yesterday to find the other sides to solve the problem.

Question:  I want to build a triangular flower bed in the corner of my yard with legs 4 ft and 3 ft.  Mulch comes in bags that cover 2 square feet, so how many bags do I need to cover the bed?  I also want to line the edge of the bed with daisy which come in 6 packs that cover 2 feet, so how many 6 packs of daisies do I need?           

Pythagorean Theorem

The Pythagorean Theorem

Objective of the lesson: Students will learn to derive and use the Pythagorean Formula to find the missing sides of a right triangle.

TEKS of the lesson- Geometry 8.C.- The student is expected to derive, extend, and use the Pythagorean Formula.

The Pythagorean Theroem states: In a right triangle, with sides (legs) a and b, and hypotenuse c, then c²=a²+b².

Note: a right triangle is a triangle with one right angle (an angle of 90°). Its hypotenuse is the side opposite the right angle.

We learned an algebraic proof using similar triangles ABC, CBX, and ACX (in the diagram):

Since corresponding parts of similar triangles are proportional, a/x=c/a or a²=cx.

b/(c-x)=c/b or b²=c²-cx or c²=cx+b².

Substituting a² for cx, we get c²=a²+b². Which is what we were trying to prove.

Next we tried some practice problems.

Example 1: If you are given a triangle with legs of length 3 and 4, what is the length of the hypotenuse?

Answer:

 3^2 + 4^2 = x^2

9 + 16 = x^2

25 = x^2

5 = x

The hypotenuse has length 5

I liked this lesson.  I think the Pythagorean theorem is going to be very useful in the future.  I wonder though- does it matter which side I call A and which side I call B?  I think it doesn't, but I just wanted to check.

Congruence vs. Similarity

Today, we learned about Congruence versus Similarity.

Objective:
Students will differentiate between congruent and similar triangles.

TEKS:
G.10.B
The student is expected to justify and apply triangle congruence
relationships.

G.11.C
The student is expected to develop, apply, and justify triangle similarity

relationships, such as right triangle ratios,
trigonometric ratios, and Pythagorean triples
using a variety of methods.

We learned that similar triangles have the same angle measures. Mr. Escalante said they have to have "corresponding angles" that are the same, which is just a fancy way of saying that each angle in one triangle has to equal an angle in the other triangle. You can use angle-angle (AA) to identify similar triangles. This works because if you know two angles, then you automatically know the third by the Angle Sum Theorem, which we already learned.

Congruent triangles have the same angle measures, but they also have the same side lengths.  Basically, a congruent triangle is a similar triangle where the sides of the first triangle are the same size as the sides of the second triangle. You can use side-side-side (SSS), side-angle-side (SAS), and angle-side-angle (ASA) to identify congruent triangles.

This lesson was kind of cool because it took the things we learned about angle sums, similarity and congruence and made them make sense. I liked how Mr. E. showed us a triangle then stretched it out without changing the angles to explain similarity.  I am still confused about why SSS is enough to make triangles congruent. Why can't the angles change when the sides are the same?

Question of the day:

House builders are working on a roof, and they want to figure out how far it will be from the peak of the roof to the bottom of the eaves.  They have a scale model of the house that is 10 inches wide from eaves to eaves. They also know that the house is 50 feet wide from eaves to eaves.  Will they use similarity or congruence to find the desired measurement? Do they have enough information to be confident? If not, what information do they need?

Triangle Congruence Theorems

Today we learned three theorems to use when finding congruent triangles.

Today’s Objective: Students will be able to determine whether two triangles are congruent and explain why.

Today’s TEKS: G.10(B) Justify and apply triangle congruence relationships.

Side Angle Side (SAS) – If two side lengths of the different triangles are equal and the angle between those sides is equal to the angle sandwiched on the other triangle, then those two triangles are congruent.

Angle Side Angle (ASA) – If two angles of the different triangles are equal to each other and the sides on each triangle between the angles are equal, then those two triangles are congruent.

Side Side Side (SSS) – If all three sides of one triangle are equivalent to all three corresponding sides of a different triangle, then those two triangles are congruent.

If the problem does not give us the measurement of the angle(s) that we need, we might have to use the angle sum theorem talked about in class earlier to find the angle's measurement.

Mr. Escalante also presented a few other congruence theorems, but told us that these 3 are the most important to remember.

Angle Sum Theorem

Today we learned about the angles in triangles and how they relate to each other. 

Today’s objective: The students will discover and apply the Interior Angle Sum Theorem and the Exterior Angle Theorem for triangles. 

Today’s TEKS: G.9.B - Formulate and test conjectures about the properties and attributes of polygons and their component parts based on explorations and concrete models.

First, we were each told to draw a triangle.  Then we learned about using a protractor to measure the interior angles in the triangles.  We all realized that the sum of the interior angles in our triangles was 180 degrees, no matter what the triangle looked like.  This is known as the Interior Angle Sum Theorem. 

Interior Angle Sum Theorem- The sum of the measures of the interior angles of a triangle is 180 degrees.

Next, we learned that the angle formed by one side of triangle with the extension of another side is called an exterior angle of the triangle.  We all drew an exterior angle on our triangles and tried to relate it to the interior angles.  We learned that the two angles that are not adjacent, or next to, the exterior angle of the triangle are called remote interior angles.  We discovered the exterior angle theorem:

Exterior Angle Theorem- The measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

I do have a question though.  How many exterior angles does a triangle have?  Is it three or six?  If someone could answer in the comments, that would be great!