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| 12 Theory slides |
| 11 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
Geometric Transformations and Similarity:
Triangles and Measurement:
Determine if the given triangles are similar using the Angle-Angle Similarity Theorem. If necessary, apply the Interior Angles Theorem to find missing angles before identifying corresponding angles.
The method of Indirect measurement uses the properties of similar triangles and proportionality to calculate distances or the dimensions of objects that are difficult to measure directly. For instance, visualize two flag poles under the sunlight; one 15 feet tall pole casts a 12-foot shadow and the other casts an 8-foot shadow. What is the height of the second flagpole?
LHS⋅15=RHS⋅15
ca⋅b=ca⋅b
Multiply
Calculate quotient
Dylan is a skater who enjoys exploring the math behind the ramps, obstacles, and shadows at a local skate park. One sunny day, he took a break under a tall tree and noticed his shadow next to the tree's shadow. This sparked his curiosity about the tree's height, and he decided to use its shadow to calculate it.
The ratio of the lengths of corresponding sides of similar triangles is always the same.
Substitute values
Add terms
LHS⋅CE=RHS⋅CE
LHS⋅79=RHS⋅79
LHS/13=RHS/13
Dylan is enjoying his time skating at the park. He drops-in on a triangular-shaped ramp. He begins his descent from a height of 15 feet, which is also the height of the ramp. After he glides down the slope for 20 feet, he realizes that he is now only 70% as high as when he started.
The ratio of the lengths of corresponding sides of similar triangles is always the same.
Substitute values
Calculate quotient
LHS⋅(20+BE)=RHS⋅(20+BE)
Distribute 0.7
LHS−0.7BE=RHS−0.7BE
Rearrange equation
LHS/0.3=RHS/0.3
Use a calculator
Later that day, Dylan returned home and realized that he still forgot about his math homework. He needs to finish it so that he can have free time on Saturday to go skateboarding again. His homework requires him to compare a pair of slope triangles that lie on the same line.
Consider that the side of each square in the grid is 1 unit. Help Dylan complete his homework.
LHS⋅BE=RHS⋅BE
LHS⋅DE=RHS⋅DE
LHS/BC=RHS/BC
LHS/DE=RHS/DE
This means that AC=3 and BC=2. Apply a similar reasoning to find the lengths of BE and DE.
riseto the horizontal change or
run. This ratio remains equal across any two slope triangles formed by that line.
At Dylan's skate park, there is a cool shelter that covers the entire skateboard track, providing shade for everyone during their skate sessions. Seeing the roof line, Dylan thinks it is the perfect chance to apply his new school knowledge about slope triangles on the same line.
With the help of a friend, Dylan measured the rise and run at various points on one of the slopes of the roof and sketched a diagram.
Now, he is curious whether the slope of the roof line remains consistent over the different areas. Help Dylan figure out the roof's slope and see if it remains the same everywhere.
Start by choosing two points on the roof line and find the slope using the Slope Formula. Next, verify that the slope is the same by selecting a different set of points.
Substitute (2,4.46) & (0,3.9)
Subtract terms
Calculate quotient
Substitute (6.25,5.65) & (4,5.05)
Subtract terms
Calculate quotient
Little Leapand
Giant Glide.Dylan draws them as triangles that look similar on paper, but
Giant Glideis just a bigger version of
Little Leap.
Little Leapramp?
Giant Glideramp?
Little Leapramp. It is formed by the legs of a triangle measuring 18.75 feet and 11.25 feet, with a hypotenuse of 21.87 feet. Add up the length of all three sides to find the perimeter of the
Little Leapramp.
Giant Glideramp can be found in a similar way. The legs of the triangle are of lengths 75 and 45 feet, respectively, while the hypotenuse has a length of 87.48 feet. Add these lengths to determine the perimeter of the triangle.
Giant Glideramp is 207.48 feet.
The slope of a line is -2.
Let's start by considering a line with a slope of -2.
We are given that the rise of a slope triangle on this line is -4. We need to determine the run of this slope triangle. To begin, we will select a lattice point on the line as our starting point.
Next, we will draw the leg of the triangle, moving down 4 units since our rise value is negative.
We know that the ratio of the rise to the run of a slope triangle is equal to the slope of the line. rise/run = slope The rise of our slope triangle is -4 and the slope of our line is - 2. This allows us to calculate the corresponding run value for our rise.
The run of this slope triangle is 2 units. We can confirm this by looking at the graph.
Let's follow a similar process as in Part A) to find the run for a slope triangle with a rise value of -5. We will select a lattice point on the line as our starting point and then move 5 units down since the rise is negative.
Since we know that the ratio of the rise to run of a slope triangle is equal to the slope of the line, we will use the slope of -2 and the rise of -5 to find the run that corresponds to this rise.
The run of this slope triangle is 2.5 units. Let's confirm this by looking at the graph.
Consider the following triangles.
We will start by finding the values of x and y by creating a pair of equations using the Interior Angles Theorem. After that, we will substitute the values of x and y into each angle expression to calculate the angle measures. Finally, we will check if corresponding angles are equal to determine if the triangles are similar. Let's start by calculating the value of x.
Consider the given diagram.
The Interior Angles Theorem states that the sum of the interior angles of a triangle equals 180^(∘). With this, we can write an equation for x.
Now, let's solve this equation to find the value of x.
We will find the value of y by using a similar reasoning. According to the Interior Angles Theorem, the sum of the interior angles of this triangle equals 180^(∘). Let's use this to write an equation for y.
Let's solve this equation to find the value of y.
We need to determine if the two triangles are similar. We will first calculate the measures of all the angles in each triangle. Since we know that x= 7.5, we can substitute 7.5 for x in the expressions 12x and 6x to find the angle measures in the left triangle. cc 12x & 6x ⇓ & ⇓ 12( 7.5) & 6( 7.5) ⇓ & ⇓ 90 & 45 Similarly, we can substitute 22.5 for y in the expressions 4y and 2y to find the angle measures in the right triangle. cc 4y & 2y ⇓ & ⇓ 4( 22.5) & 2( 22.5) ⇓ & ⇓ 90 & 45 Once we have found the angle measures for both triangles, we can add this information into the diagram.
We can see that the angles in the left triangle are congruent to corresponding angles in the right triangle. Therefore, we can conclude that the two triangles are similar.