a. Identify whether or not the two triangles below are similar. If so, create a similarity statement and give a reason.
b. If BC=21, EC=18, and DC=14, find AE.
There are a few ways to approach part b. Hopefully you can get to an answer of AE=9. We discussed the various ways to arrive at this answer which led to the key ideas of what is called the Triangle Proportionality Theorem.
Hints for Springboard problems for PBA:
p. 241: 4.) Function means (x,y)-> so show what happens to x and y according to the rule.
9.) Create a list of numbers that you think would work as scale factors for reductions. Where are all of those numbers. Do not use words like “fractions” or “decimals” in your answer. Consider what a scale factor of 1 would do to a shape.
10.) Same hint as above.
13.) a.) Pay close attention to the ratio given: What is the difference between the top and bottom?
The left ratio needs to be related to the right one shown in numbers.
Read the proportion as “Prime to not prime is equivalent to 5 to 2,” so what part has to do with the 5, versus the 2? Is the image or preimage the smaller part of the ratio?
b.) Rearrange the given proportion by multiplying each side by “RS” to get R’S’=5/2*RS. So you can see that 5/2 is actually the scale factor being used for the dilation. Also notice that the coordinates of R’ and R are the same, so nothing happened to it during the dilation, meaning those coordinates must be the center of dilation. Now you know the scale factor and center of dilation, so you can find the other coordinates.
5.a. Use the correct phrases/keywords when talking about sides in congruent figures versus in similar figures.
5.b. Look in your notes!
6.a.b. Same hints as 5!
- Be careful to not just use “S” and “A” just because you see them on a shape. “A” means that the same angle is actually present, in the same structure, in each shape. Each “S” means that a pair of sides is proportional to another pair of sides, so you have to show the proportion and show that it actually works out correctly.
- What does isosceles mean? What do you know about the angles of an isosceles triangle? What do you know about right triangles? What must the exact measures be in this case then? What does that say about any two isosceles right triangles? What do they have in common?
- Similar to above, what do you know about the angles? What does that say about any two equilateral triangles?
- Don’t forget about Vertical Angles Theorem, and to also actually show and check the proportions.
- Please be sure to use all letters necessary when naming angles on the diagram (for example, there is no angle B)
- Use the vertical angles theorem and the triangle sum theorem (all 3 angles add up to equal 180) Then you should see some common information between the two triangles allowing you to use a structure.
- To fill out the proportion- what goes beneath the 65? What corresponds with the 65 in the other triangle? Same question for the other side of the proportion.
- What would Jorge need to know before setting up his proportion? Why is this an issue?
4.b. Show the two triangles are similar first! Check proportions of corresponding sides, then create a proportion to solve for the missing side.
5. Be careful about which sides correspond- use the angle placement to double check, and then create your proportion
- Use the parallel lines and the corresponding angles postulate.