Question

geometry midterm exam
name
score:

1. the points a(-9, 2), b(-2, 2) and c(-2, 7), are plotted and connected to form

△abc.
a. reflect (flip) △abc across the line y = x. label this △abc. **hint: use your
green transformation toolkit**
a ____ b __ c ____
b. find the slope and length of
overline{ac}.
slope: _____ length: _____

1. for each diagram, set up an equation and solve for x. name what angle relationships you used to solve the

problem. hint: use your purple angle relationship toolkit
a.
b.
x = _____ relationship: _____
x = _____ relationship: _____

1. erin, who is 1.6 meters tall, casts a shadow that is 1.25 meters long. her shadow extends to the end of a trees

shadow when she stands 4.75 meters from the tree. what is the height of the tree? draw a diagram to support
your answer. hint: draw as two separate triangles

1. determine the triangles are similar. if similar, create a flowchart to prove the similarity. if not, state why they

are not similar. note: figures may not be drawn to scale.

Explanation:

Problem 1a: Reflect $\triangle ABC$ across $y = x$

Step1: Recall reflection over $y = x$

The rule for reflecting a point $(x, y)$ over $y = x$ is $(x, y) \to (y, x)$.

Step2: Apply to each vertex

• For $A(-9, 2)$: Swap $x$ and $y$ → $A'(2, -9)$.
• For $B(-2, 2)$: Swap $x$ and $y$ → $B'(2, -2)$.
• For $C(-2, 7)$: Swap $x$ and $y$ → $C'(7, -2)$.

Problem 1b: Slope and Length of $\overline{AC}$

Step1: Slope formula

Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$. For $A(-9, 2)$ and $C(-2, 7)$:
$m = \frac{7 - 2}{-2 - (-9)} = \frac{5}{7}$.

Step2: Distance formula

Length $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
$d = \sqrt{(-2 - (-9))^2 + (7 - 2)^2} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74}$.

Problem 2a: Solve for $x$ (Angle Relationship)

Step1: Identify relationship

The angles are supplementary (form a linear pair), so $6x - 24 + 2x + 12 = 180$.

Step2: Solve the equation

Combine like terms: $8x - 12 = 180$.
Add 12: $8x = 192$.
Divide by 8: $x = 24$.

Problem 2b: Solve for $x$ (Angle Relationship)

Answer:

s:
1a. $A'(2, -9)$, $B'(2, -2)$, $C'(7, -2)$
1b. Slope: $\boldsymbol{\frac{5}{7}}$, Length: $\boldsymbol{\sqrt{74}}$
2a. $x = \boldsymbol{24}$, Relationship: Supplementary Angles
2b. $x = \boldsymbol{21}$, Relationship: Alternate Interior Angles

1. Tree height: $\boldsymbol{7.68}$ meters
2. Triangles are similar by SAS similarity ( $\angle Q = \angle X$, $\frac{PQ}{XY} = \frac{DQ}{XZ} = \frac{5}{4}$ )