SLO 2.4: Proportionality, Ratios and the Scale Factor

SLO 2.4: Proportionality, Ratios and the Scale Factor

The ratio of corresponding side lengths IS the scale factor in similar triangles

Cross Multiply to Solve Proportions

• Set up your Side Length (SL) proportions:

•  $\frac{image\ SL\ 1}{preimage\ SL\ 1}=\frac{image\ SL\ 2}{preimage\ SL\ 2}$  

• Cross multiply and solve

Fill in the Blank

Solve this proportion (can you butterfly?)

 $\frac{3}{8}=\frac{x}{24}$  

(Type just the number)

This Proportion IS the Scale Factor

• Sometimes you can immediately recognize the scale factor and proportion the answer without cross-multiplying

• In this example compare the denominators

• 24 is 3 times as big as 8

   $\left(k=3\right)$  

• Therefore x must by 3 times as big as 3

Fill in the Blank

Solve this proportion (can you butterfly?)

 $\frac{5}{7}=\frac{25}{y}$  

(Type just the number)

Fill in the Blank

Solve this proportion (can you butterfly?)

 $\frac{5}{t}=\frac{t}{45}$  

(Type just the number)

Fill in the Blank

Find the indicated length (x) for this pair of similar figures.

(Type only the number)

Multiple Choice

Use the proportional relationships to find the values of  $x,\ y,\ z$   for this pair of similar figures.

•  $x=12.5;\ y=3.2;\ z=6$  

•  $x=12;\ y=4;\ z=5$  

•  $x=5;\ y=8;\ z=15$  

Multiple Select

Which of the following figures are similar to Figure X? (check all that apply)

• a. (9/12/15)

• b. (15/80/100)

• c. (3/4/9)

• d. (10.5/14/17.5)

Pythagorean Triplets

• Tie 12 evenly spaced knots into a rope to guarantee your corner is a right angle

•  $3^2+4^2=5^2$  

• Apply any scale factor  $k=10;\ k=3.5$  

• There are lots of Pythagorean Triplets but 3-4-5 is the most common by far  $\left(5\cdot12\cdot13\right);\ \left(8\cdot15\cdot17\right);\ \left(7\cdot24\cdot25\right)$  

Open Ended

For this problem you are given that $\overline{AB}$  is parallel to $\overline{DE}$ and asked to find the length of  $\overline{CD}$  .  Without knowing that those two lines are parallel you could NOT solve this problem because you are NOT given that the two triangles are similar.  How can you prove the two triangles are similar?

Fill in the Blank

In the diagram, $\overline{AB}$ is parallel to $\overline{DE}$ . AB = 20 inches, DE = 12 inches, and BC = 15 inches. What is the length of $\overline{DC}$ ? (Type only the number)

Fill in the Blank

A rectangle has a length of 4 inches and a perimeter of 14 inches. What is the perimeter of a similar rectangle with a width of 9 inches? (Type the number only)

Multiple Choice

Brandon wants to reduce a figure that is 9 inches tall and 16 inches wide so that it will fit on a 9-inch-by-12-inch piece of paper. If he reduces the figure proportionally, what is the maximum size the reduced figure could measure?

• $12\ \times21\ \frac{1}{3}\ \ inches$

• $9\times12\ \ inches$

• $5\ \frac{1}{16}\times9\ \ inches$

• $6\ \frac{3}{4}\times12\ \ inches$

Fill in the Blank

The sail shown has a horizontal stripe parallel to the base of the sail. What is the distance, x, from the bottom of the sail to the stripe? Hint: the height of the entire sail is (x + 10). Type in only the number as your answer)

Fill in the Blank

Knowing that triangle TSR is similar to triangle TNM (because all the corresponding angles are congruent) , find the length of x. (Type the number only)

Multiple Choice

A triangle with vertices (1, 2), (5, 2), and (3, -2) is shown, giving the triangle a base length of 4 units and a height (altitude) of 4 units.

Which triangle below is similar to the figure above?

Multiple Choice

Use the information in each diagram to determine which pairs of polygons are similar. (Choose only one)

Multiple Choice

A certain parallelogram has the dimensions shown. Which set of dimensions would produce a similar figure?

• 17.6 cm, 88 cm

• 70.4 cm, 176 cm

• 105.6 cm, 132 cm

• 140.8 cm, 220 cm

Multiple Choice

If SQ = 6 units, QT = 24 units, and the perimeter of triangle SQP (the little triangle) is 20 units, what is the perimeter of triangle STR (the big triangle) ?

(Hint: The length of ST is 6 + 24 = 30)

• 80 units

• 100 units

• 320 units

• 500 units

Fill in the Blank

To estimate the height of her school’s gym, Nicole sights the top of the gym wall in a mirror that she has placed on the ground. The mirror is 3.6 meters from the base of the gym wall. Nicole is standing 0.5 meter from the mirror, and her height is about 1.8 meters. What is the height of the gym wall? (Type in only the number to 2 decimal places).

Missing Information

• Two triangles can be proven similar by:

• AA or AAA (same thing)

• SAS (two sides proportional, one congruent angle)

• SSS (all sides proportional)

• What information is missing from the last slide (mirror problem)? You can't prove these triangles similar unless you know...