Question

1. for the diagram below,

a. make a flowchart showing these triangles are similar.
b. find the values of x and y.
$x = \underline{\quad\quad}$
$y = \underline{\quad\quad}$

Explanation:

Part a:

1. Start with given right angles: $\angle PVW = \angle PQR = 90^\circ$
2. Identify common angle: $\angle QPR$ is shared by $\triangle PVW$ and $\triangle PQR$
3. Apply AA (Angle-Angle) Similarity Criterion: If two angles of one triangle match two angles of another, triangles are similar.
4. Conclude similarity: $\triangle PVW \sim \triangle PQR$

Flowchart structure:

Start
  |
  V
Given: ∠PVW = 90°, ∠PQR = 90°
  |
  V
∠PVW ≅ ∠PQR (Right angles are congruent)
  |
  V
∠QPR ≅ ∠QPR (Reflexive Property)
  |
  V
△PVW ~ △PQR (AA Similarity Criterion)
  |
  V
End

---

Part b:

Step1: Set up proportion for x

Corresponding sides of similar triangles are proportional: $\frac{PV}{PQ} = \frac{PW}{PR}$
Substitute values: $\frac{9}{18} = \frac{x}{x+22}$

Step2: Solve for x

Cross-multiply to eliminate fractions:
$9(x+22) = 18x$
$9x + 198 = 18x$
$198 = 18x - 9x$
$9x = 198$
$x = \frac{198}{9} = 22$

Step3: Set up proportion for y

Use proportional sides: $\frac{PV}{PQ} = \frac{VW}{QR}$
Substitute values: $\frac{9}{18} = \frac{6}{y}$

Step4: Solve for y

Cross-multiply:
$9y = 18 \times 6$
$9y = 108$
$y = \frac{108}{9} = 12$

Answer:

$x = 22$
$y = 12$