Triangles

Triangles: Congruence and Similarity

Day 2

Before we can discuss congruency and similarity, we need to talk about what it means to be corresponding.

Corresponding parts are those that appear in the same place in two similar situations

*Be careful when the figures appear to be located in a different way.

Congruent

• When two figures are exactly the same size and shape even if the figure was rotated

Similar

• when corresponding angles are congruent and the corresponding sides are in proportion. There must be a scale factor that applies to all the side lengths.

Congruent or Similar

Congruent or Similar

Congruent or Similar

There is more to similarity

• You want to check that the angles are the same in both figures. Make sure to see if there is no more information you can add from your knowledge

• Make sure all corresponding sides have the same ratio and choose the sides strategically.

• With word problems, DRAW THE FIGURE!!

Pythagorean Theorem

Math Antics

​https://youtu.be/WqhlG3Vakw8

Some gossip...

- Rumor is Pythagoras didn't actually discover it

- The concept appears in a 4000 year old Babylonian Tablet (Plimpton 322). Pythagoras "officially" made it a theorem in 1900 B.C.

We use the pythagorean theorem to find a missing side length or to prove if the triangle is or isn't a right triangle

Side Length "names"

"Leg" refers to the sides of the triangle that create the right angle

"Hypotenuse" refers to the longest side of the triangle and it's ALWAYS across from the 90 degree angle.

What does the theorem actually say?

The sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse (the side opposite the right angle)

- Encyclopedia Britannica

Practice

Now that we have practiced using the theorem to prove if the triangle is a right triangle, let's work with examples where we have to find the missing side.

We've also have word problems!

Remember for problems describing a figure, drawing the figure is very helpful

​Day 4: Practice Problems

• Types of Triangles

• Angles Sum

• Pythagorean Theorem

• Congruency and Similarity​

​

​Practice

Fill in the Blanks

The lengths of two sides of a right triangle are given. Find the length of the remaining side to the nearest tenth unit.

leg a: 8 in

leg b: 8in

Hypotenuse c: ? in

Fill in the Blanks

The lengths of two sides of a right triangle are given. Find the length of the remaining side to the nearest tenth unit.

leg a: ? m

leg b: 3 m

Hypotenuse c: 6 m

Multiple Choice

On a coordinate plane, points A, B, and C can be connected to form a right triangle.

What is the distance from A to C, to the nearest tenth unit?

• 5.2

• 6.7

• 8.4

• 10.1

• 11.3

Multiple Choice

Jan has built a rectangular frame out of wood to use for the bottom of a platform. He wants to add a diagonal brace as shown in the drawing.

What will the length of the brace be, to the nearest tenth of a foot?

• 16.0

• 14.2

• 13.7

• 12.8

• 12.1

Fill in the Blanks

If a tree casts a 24ft shadow at the same time that a yardstick casts a 2ft shadow, find the height of the tree.

Fill in the Blanks

Find the height of the giraffe in the diagram below.

Fill in the Blanks

On level ground, the base of a tree is 20ft from the bottom of a 48ft flagpole. The tree is shorter than the pole. At a certain time, their shadows end at the same point 60ft from the base of the flagpole. How tall is the tree?

The bottom of a ladder must be placed 3 feet from a wall.  The ladder is 12 feet long.  How far above the ground does the ladder touch the wall?

A soccer field is a rectangle 90 meters wide and 120 meters long. The coach asks players to run from one corner to the corner diagonally across the field. How far do the players run?

How far from the base of the house do you need to place a 15’ ladder so that it exactly reaches the top of a 12’ wall?

How far from the base of the house do you need to place a 15’ ladder so that it exactly reaches the top of a 12’ wall?