Lesson 5

Splitting Triangle Sides with Dilation, Part 1

  • Let’s draw segments connecting midpoints of the sides of triangles.

Problem 1

What is the measure of angle \(A’B’C\)?

Triangle A B C with segment A’ B’ drawn. A A’ is 2 and A’ C is 4. B B’ is 3 and B’ C is 6. Angle B is 40 degrees.
A:

\(20^\circ\)

B:

\(40^\circ\)

C:

\(60^\circ\)

D:

\(80^\circ\)

Problem 2

Triangle \(DEF\) is formed by connecting the midpoints of the sides of triangle \(ABC\). The lengths of the sides of \(DEF\) are shown. What is the length of \(AB\)?

Triangles A B C and D E F. D is the midpoint of segment A B. E is the midpoint of segment B C. F is the midpoint of segment A C. Line D E has length 2, Line E F has length 3, Line D F has line 4.  

Problem 3

Angle \(ABC\) is taken by a dilation with center \(P\) and scale factor \(\frac13\) to angle \(A’B’C’\). The measure of angle \(ABC\) is \(21^\circ\). What is the measure of angle \(A’B’C’\)?

(From Unit 3, Lesson 4.)

Problem 4

Draw 2 lines that could be the image of line \(m\) by a dilation. Label the lines \(n\) and \(p\).

2 intersecting lines labeled o and m.
(From Unit 3, Lesson 4.)

Problem 5

Is it possible for polygon \(ABCDE\) to be dilated to figure \(VWXYZ\)? Explain your reasoning.  

Polygon V W X Y Z. Side V W labeld 8. Side W X labeled 10. Side X Y labeled 6. Side Y Z labeled 8. Side Z V labeled 12.  
Polygon A B C D E. Side A B labeled 4. Side B C labeled 5. Side C D labeled 3. Side D E labeled 4. Side E A labeled 10.  

 

(From Unit 3, Lesson 3.)

Problem 6

Triangle \(XYZ\) is scaled and the image is \(X'Y'Z'\). Write 2 equations that could be used to solve for \(a\). 

Triangles X Y Z and X prime Y prime Z prime. Side X Y is 8 and side X Z is 5. Side X prime Y prime is 3 and side X prime Z prime is a.
(From Unit 3, Lesson 2.)

Problem 7

  1. Lin is using the diagram to prove the statement, “If a parallelogram has one right angle, it is a rectangle.” Given that \(EFGH\) is a parallelogram and angle \(HEF\) is a right angle, write a statement that will help prove angle \(FGH\) is also a right angle.
  2. Han then states that the 2 triangles created by diagonal \(EG\) must be congruent. Help Han write a proof that triangle \(EHG\) is congruent to triangle \(GFE\).
Quadrilateral EFGH with angle HEF as a right angle. Line segments EG and HF intersect at point D.
(From Unit 2, Lesson 12.)