Geometry Basics: Chapter Test

A chord is any straight segment whose two ends are points of the circle. A diameter has both its ends on the circle and happens to pass through the center as well, so it meets that description and is the longest chord of all.

$\text{every diameter is a chord}, \qquad \text{not every chord is a diameter}$

The other three names each fail on where their ends sit: a radius has one end at the center, and an arc is not straight at all but a piece of the boundary itself.

The two angles sit side by side and their outer sides form a straight line, so they are a linear pair and add to a straight angle.

$43^\circ + x = 180^\circ \quad\Longrightarrow\quad x = 180^\circ - 43^\circ = 137^\circ$

A check: $43^\circ + 137^\circ = 180^\circ$, which is the straight line the ray was drawn on.

An ordered pair always reports the sideways position first and the up-or-down position second, whatever order the moves happened in. Left is negative and down is negative.

$x = -8, \qquad y = -3 \quad\Longrightarrow\quad (-8, -3)$

The two moves can be made in either order and land in the same place, because one changes only the first coordinate and the other only the second.

Two copies of a triangle fit together into a parallelogram on the same base and the same perpendicular height, so a triangle covers half of that.

$A = \tfrac{1}{2} \times b \times h = \tfrac{1}{2} \times 38 \times 21 = 399 \text{ cm}^2$

The $30$ cm sloping edge is one of the lengths the boundary is made of, and it never enters the area.

The radius is given, so use the form of the circumference rule built on the radius.

$C = 2\pi r \approx 2 \times 3.14 \times 23 = 144.44 \text{ cm}$

The diameter route agrees: $d = 46$ cm, and $3.14 \times 46 = 144.44$. The answer is approximate because $\pi$ was rounded.

Angles directly opposite each other at a crossing are vertical angles, and vertical angles are equal, so the two expressions may be set equal.

$5x + 9 = 8x - 27 \quad\Longrightarrow\quad 36 = 3x \quad\Longrightarrow\quad x = 12$

Substituting back into either expression gives the measure: $5(12) + 9 = 69$ and $8(12) - 27 = 69$. Both agree, which is the check that $x$ was solved correctly.

Every length has to be in one unit before an area is computed, and the answer is wanted in square meters, so convert the side first: $75 \text{ cm} = 0.75$ m.

$A = 2.4 \times 0.75 = 1.8 \text{ m}^2$

The centimeter route agrees once the square unit is converted properly: $240 \times 75 = 18000 \text{ cm}^2$, and $18000 \div 10{,}000 = 1.8 \text{ m}^2$, because a square meter is $100$ cm by $100$ cm.

The path has exactly one end, at $J$, and runs forever the other way, which is what a ray is. A ray is always named from its own endpoint first.

$\text{one endpoint} + \text{one endless direction} = \overrightarrow{JK}$

The count of endpoints separates the three straight objects: a line has none, a ray has one, a segment has two.

A prism's volume is the area of one base multiplied by the distance to the identical base opposite, so find the triangle's area first.

$B = \tfrac{1}{2} \times 9 \times 6 = 27 \text{ cm}^2, \qquad V = B \times h = 27 \times 11 = 297 \text{ cm}^3$

The base area is in square centimeters and the length adds a third factor of centimeters, which is why the volume is cubic.

Angles in matching positions at the two crossings are corresponding angles, and the chevrons mark the crossed lines as parallel, so the two are equal.

$8x - 15 = 5x + 27 \quad\Longrightarrow\quad 3x = 42 \quad\Longrightarrow\quad x = 14$

Each angle is then $8(14) - 15 = 97$, and the other expression agrees at $5(14) + 27 = 97$.

Cut the T along the underside of the bar, which leaves two rectangles that do not overlap, and add their areas.

$A = (25 \times 8) + (9 \times 19) = 200 + 171 = 371 \text{ cm}^2$

The stem is $19$ cm tall measured from where it meets the bar, so the plaque stands $8 + 19 = 27$ cm tall in all, and the rectangle it fits inside covers $25 \times 27 = 675 \text{ cm}^2$, more than the plaque itself.

Split the window into the rectangle and the semicircle, find each area, and add. The semicircle's straight edge is a diameter, so halve it for the radius: $r = 17$ cm.

$A_{\text{rect}} = 34 \times 50 = 1700, \qquad A_{\text{semi}} = \tfrac{1}{2} \times 3.14 \times 17^2 = 453.73$

Adding the two non-overlapping pieces gives $1700 + 453.73 = 2153.73 \text{ cm}^2$.

A cylinder's volume is its base area times its height, so the height is the volume divided by the base area. Find that base area first.

$B = \pi r^2 \approx 3.14 \times 18^2 = 1017.36 \text{ cm}^2$

Now undo the multiplication.

$h = \frac{20347.2}{1017.36} = 20 \text{ cm}$

A check forward: $1017.36 \times 20 = 20347.2$.

Quadrant IV is the lower right, so a point there is to the right of the y-axis and below the x-axis: the first coordinate is positive and the second negative.

$a > 0, \; b < 0 \quad\Longrightarrow\quad (b, a) \text{ has first coordinate } b < 0 \text{ and second coordinate } a > 0$

A negative first coordinate puts the new point left of the y-axis and a positive second coordinate puts it above the x-axis, which is the upper left, Quadrant II.

The trapezoid rule multiplies the average of the two parallel sides by the height, so dividing the area by the height undoes the last step and leaves that average.

$\frac{336}{14} = 24 = \frac{19 + b}{2} \quad\Longrightarrow\quad 19 + b = 48 \quad\Longrightarrow\quad b = 29 \text{ cm}$

A check forward: $\frac{19 + 29}{2} \times 14 = 24 \times 14 = 336$.

The answer is wanted in cubic meters, so put all three measurements in meters before multiplying: $80 \text{ cm} = 0.8$ m and $50 \text{ cm} = 0.5$ m.

$V = 1.2 \times 0.8 \times 0.5 = 0.48 \text{ m}^3$

Working in centimeters agrees once the cubic unit is converted properly: $120 \times 80 \times 50 = 480{,}000 \text{ cm}^3$, and a cubic meter is $100 \times 100 \times 100 = 1{,}000{,}000$ cubic centimeters, so that is $0.48 \text{ m}^3$.

Alternate interior angles are equal when the crossed lines are parallel, and the chevrons say they are, so set the two expressions equal.

$6x + 33 = 4x + 69 \quad\Longrightarrow\quad 2x = 36 \quad\Longrightarrow\quad x = 18$

Each of the two angles is then $6(18) + 33 = 141^\circ$. The angle asked for sits beside that one on a straight line, so it is its supplement.

$180^\circ - 141^\circ = 39^\circ$

Every circle has the same circumference-to-diameter ratio, so the first wheel's two measurements fix that ratio once and for all.

$\frac{182.12}{58} = 3.14$

The second wheel must give the same ratio, so its rim is that number of times its own width.

$C = 3.14 \times 87 = 273.18 \text{ cm}$

Scaling gives the same figure without ever naming the ratio: $87$ is $1.5$ times $58$, and $1.5 \times 182.12 = 273.18$.

Unfold the box into its net. The six faces come in three matching pairs, so work out one of each kind and double the total.

$34 \times 18 = 612, \qquad 34 \times 12 = 408, \qquad 18 \times 12 = 216$

Adding the three and doubling gives the cardboard.

$S = 2(612 + 408 + 216) = 2 \times 1236 = 2472 \text{ cm}^2$

The unit is square centimeters, because every piece of a net is a flat area.

The two given corners share the column $x = -4$, so the side joining them is vertical and its length is the gap between their heights.

$w = |6 - (-5)| = 11$

The area rule for a rectangle is a plain product, so dividing by that side recovers the horizontal one.

$11 \times l = 209 \quad\Longrightarrow\quad l = \frac{209}{11} = 19$

Stepping $19$ units to the right from $x = -4$ lands the far corners in the column

$-4 + 19 = 15.$

A check: the corners $(15, 6)$ and $(15, -5)$ give $|15 - (-4)| \times 11 = 19 \times 11 = 209$.