Question

1. (lsm) find ( m overarc{dh} + m overarc{fi} ) of the circle below. (hint: first, find ( x ))

a. ( 270^circ )
b. ( 90^circ )
c. ( 135^circ )
d. ( 210^circ )
e. ( 120^circ )

1. (l109) identify the relationship between figures a and d in the coordinate plane below.

a. translation
b. rotation
c. reflection
d. construction
e. similarity

1. (l109) identify the relationship between figures b and d in the coordinate plane above.

a. translation
b. rotation
c. reflection
d. construction
e. similarity

1. (l97) the volume of the square pyramid below is 490, and its altitude is 30.

find the length of edge ( s ).
a. 4.04
b. 16.3
c. 12.12
d. 7.00
e. 8.56

Explanation:

Question 18

Step1: Use secant-secant power theorem

For a point outside a circle, the product of the lengths of the entire secant segment and its external segment is equal for both secants:
$$GF \cdot GE = GH \cdot GD$$
Substitute values: $12 \cdot (12 + x) = 9 \cdot (9 + x + 16)$

Step2: Simplify and solve for x

Expand both sides:
$$144 + 12x = 9 \cdot (25 + x)$$
$$144 + 12x = 225 + 9x$$
Subtract $9x$ and $144$:
$$3x = 81$$
$$x = 27$$

Step3: Identify arcs as straight angles

$\overarc{DH}$ is the arc corresponding to chord $DE + EH = x + 9$, and $\overarc{FI}$ corresponds to chord $FE + EI = x + 16$. Since $\overarc{DH}$ and $\overarc{FI}$ are arcs that form a pair of supplementary arcs with the remaining circle? No, correction: The arcs $\overarc{DH}$ and $\overarc{FI}$ are the arcs intercepted by the chords, but since $x=27$, the chords $ID = x+16=43$, $FH=x+9=36$, but the key is that $\overarc{DH} + \overarc{FI} = 180^\circ$? No, correction: When two chords intersect, the measure of vertical angles relate to arcs, but we need the sum of the arcs. Wait, no: The secant segments give us that the arcs $\overarc{DH}$ and $\overarc{FI}$ are such that their sum is $270^\circ$? No, correction: The sum of the measures of $\overarc{DH}$ and $\overarc{FI}$ is equal to $270^\circ$? Wait, no, let's use the fact that the angle at G is right, so $\angle G = 90^\circ$, and $\angle G = \frac{1}{2}(m\overarc{DH} - m\overarc{FI})$? No, $\angle G$ is formed by two secants, so $\angle G = \frac{1}{2}(m\overarc{DH} - m\overarc{FI})$. We know $\angle G=90^\circ$, so $90 = \frac{1}{2}(m\overarc{DH} - m\overarc{FI})$, so $m\overarc{DH} - m\overarc{FI}=180$. Also, the total circle is $360^\circ$, so $m\overarc{DH} + m\overarc{FI} + m\overarc{HF} + m\overarc{ID}=360$. But $\overarc{HF}$ and $\overarc{ID}$ are supplementary to the other arcs? No, wait, we found $x=27$, so the chords: $FE=27$, $EI=16$, $DE=27$, $EH=9$. The arcs $\overarc{FI}$ and $\overarc{DH}$: the sum $m\overarc{DH} + m\overarc{FI} = 270^\circ$ (since the remaining arcs sum to $90^\circ$, and $\angle G=90^\circ$ is half the difference of the intercepted arcs, so solving gives the sum as $270^\circ$).

Figure A and Figure D are oriented such that one can be mapped to the other by rotating 180 degrees around the origin; this matches the definition of a rotation.

Figure B and Figure D are mirror images across the x-axis; this transformation is a reflection, where each point is mapped to its mirror counterpart over a line.

Answer:

A. $270^\circ$

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Question 19