Tüm alıştırma soruları

Chords $AB$ and $CD$ intersect at point $E$ inside a circle. If $AE = 6$, $EB = 8$, and the total length of chord $CD$ is $16$, what is the length of the shorter segment of chord $CD$?

In triangle $ABC$, the length of side $a$ is $5$ centimeters, the length of side $b$ is $8$ centimeters, and the measure of angle $C$ is $60^\circ$. What is the length, in centimeters, of side $c$?

A coordinate grid contains a right triangle, $XYZ$, with the right angle located at vertex $Z$. The horizontal leg $XZ$ has a length of $24$ units, and the vertical leg $YZ$ has a length of $7$ units. What is the value of $\cos(X)$?

For an angle $\theta$ such that $\frac{\pi}{2} < \theta < \pi$, if $\frac{\cos\theta}{1 - \sin\theta} = -3$, what is the value of $\sin\theta - \tan\theta$?

### Origin of the Moon

Scientists debate how the Moon was formed. Two primary hypotheses have been proposed:

Hypothesis 1
The Moon formed from the debris of a collision between the early Earth and a Mars-sized protoplanet. The impact ejected mostly the silicate-rich outer mantles of both bodies into orbit, where they accreted to form the Moon. Because the iron cores of both bodies merged to remain with Earth, the Moon was left with a very small iron core.

Hypothesis 2
The Moon formed independently in a different region of the solar nebula and was later captured by Earth's gravitational field during a close planetary flyby. This explains why the Moon has a much lower bulk density than Earth, as it formed from materials in a region of the nebula that was naturally depleted of iron.

New Evidence
Analysis of lunar rock samples reveals that the oxygen isotope ratios ($^{17}\text{O}/^{16}\text{O}$ and $^{18}\text{O}/^{16}\text{O}$) of lunar rocks are identical to those of Earth's mantle. Bodies that form in different regions of the solar nebula generally possess distinct, unique oxygen isotope signatures.

Based on this information, how does the new evidence affect the two hypotheses?

In the standard $(x, y)$ coordinate plane, a line passes through the points $(3, -1)$ and $(-1, y)$. If the slope of the line is $-2$, what is the value of $y$?

In the figure, point $B$ lies on the line segment $AC$, and segment $BD$ is perpendicular to $AC$. Triangle $ABD$ is a right triangle with hypotenuse $AD = 8$ and $\angle ADB = 30^\circ$. If the length of segment $BC$ is $11$, what is the length of segment $CD$?

A straight skateboard ramp has a vertical height of $9$ feet and a horizontal length of $12$ feet along the ground. The vertical height and horizontal length meet at a right angle. What is the sine of the angle that the ramp makes with the ground?

In the early summer of $1930$, off the coast of Nonsuch Island, Bermuda, a bizarre vessel was lowered from the deck of the barge Ready into the sapphire waters of the Atlantic. This was the Bathysphere, a spherical steel deep-sea submersible designed by Otis Barton and championed by the renowned naturalist William Beebe. At a time when deep-sea exploration was limited to dredging nets that pulled up crushed, lifeless specimens, Beebe sought to observe marine organisms in their native, undisturbed habitats. The design of the Bathysphere was a marvel of utilitarian engineering. Barton realized that only a sphere could uniformly distribute the crushing hydrostatic pressure of the abyss, which increases by approximately $1$ atmosphere for every $33$ feet of depth. Cast by the Watson-Stillman Hydraulic Machinery Company, the sphere measured exactly $4$ feet $9$ inches in diameter and possessed steel walls $1\frac{1}{4}$ inches thick. Its weight, a formidable $5,400$ pounds, required a heavy-duty winch and a specialized steel cable capable of supporting several tons without twisting.

To allow Beebe to look out into the darkness, Barton installed $3$ window ports, though only $2$ were eventually fitted with windows while the third was plugged with steel. These windows were not made of standard glass, which would have shattered under the intense pressure, but of fused quartz cylinders. Each cylinder was $8$ inches in diameter and $3$ inches thick, ground and polished to optical perfection by the General Electric Company. To seal these windows against the steel frame, Barton utilized a mixture of red lead and putty, baking the assemblies in an oven to harden the seal. The interior of the sphere was cramped and hostile. Beebe and Barton had to crawl through a circular hatch just $14$ inches in diameter, which was then secured by a heavy steel lid fastened with $10$ large brass bolts. Inside, the researchers sat on the cold steel floor, surrounded by essential life-support equipment.

Oxygen was supplied from $2$ high-pressure cylinders, each containing oxygen at $150$ atmospheres, which was released through a reducing valve at a constant rate of $2$ liters per minute—sufficient for both men for up to $8$ hours. To prevent the accumulation of toxic gases, $2$ wire mesh baskets were suspended from the ceiling. One basket contained $3$ pounds of soda lime to absorb carbon dioxide, while the other held $3$ pounds of calcium chloride to remove moisture exhaled by the divers. A single, custom-designed cable connected the Bathysphere to the surface. This cable housed both a telephone line and a power wire that fed a $250$-watt spotlight. On the deck of the Ready, Gloria Hollister, a research associate and an accomplished zoologist in her own right, sat with a telephone headset, recording Beebe’s breathless, real-time descriptions as they descended.

On their historic dive of June 11, $1930$, the pair reached a depth of $800$ feet, far deeper than any human had survived in a submersible before. Over the next $4$ years, they pushed the limits of the vessel, culminating in a record-breaking dive to $3,028$ feet on August 15, $1934$. In the pitch-black depths, illuminated only by the faint glow of the spotlight and the bioluminescence of passing creatures, Beebe witnessed a world never before seen by human eyes. He observed species that were entirely new to science, such as the *Bathysphaera intacta* (the giant dragonfish) and the *Bathysidus pentagrammus* (the five-lined constellation fish). Because these observations were fleeting and could not be verified by physical specimens, some contemporary ichthyologists reacted with intense skepticism. They argued that the optical distortions of the thick quartz windows, combined with the psychological effects of sensory deprivation and nitrogen narcosis, had caused Beebe to misidentify known species or imagine entirely new ones. Nevertheless, Beebe’s meticulous notes, preserved through his telephone conversations with Hollister, laid the groundwork for modern deep-sea biology.

Based on the passage, is the statement that the Watson-Stillman Hydraulic Machinery Company ground and polished the Bathysphere’s $3$-inch-thick quartz window cylinders true or false?

A point on the unit circle starts at $(1, 0)$ and undergoes a counterclockwise rotation of $570^\circ$ about the origin. What is the radian measure of the angle in the interval $[0, 2\pi)$ that corresponds to the point's final position?

In the standard $(x, y)$ coordinate plane, a parallelogram $ABCD$ has vertices $A(-6, -3)$, $B(1, -3)$, $C(6, 2)$, and $D(-1, 2)$. What is the length of the diagonal $AC$?

An angle $\theta$ in standard position measures $-750^\circ$. Let $\phi$ be the coterminal angle of $\theta$ such that $0 \le \phi < 2\pi$ radians. If $\phi = \frac{a\pi}{b}$, where $a$ and $b$ are positive integers with no common factors, what is the value of $a + b$?

A line graphed on the standard $(x, y)$ coordinate plane has a slope of $0.5$ and passes through the points $(p, 3)$ and $(7, 2p - 1)$. What is the value of $p$?

Deep-sea hydrothermal vents, first discovered in $1977$, are fissures on the seafloor that release geothermally heated water. Because sunlight cannot penetrate the ocean depths where these vents exist, photosynthesis is impossible. Instead, these ecosystems rely on chemosynthesis, a process in which specialized microbes utilize chemical compounds, primarily hydrogen sulfide, to produce energy. The sudden emergence of these mineral-rich hot springs results in a highly localized deposition of copper, iron, and zinc sulfides onto the ocean floor, forming towering chimneys known as "black smokers." As the superheated water, reaching temperatures up to $400$ degrees Celsius, meets the near-freezing ocean water, the dissolved minerals precipitate out of solution immediately. This rapid precipitation not only constructs the physical structures of the chimneys but also establishes a dense chemical gradient. Crucially, the high concentration of hydrogen sulfide emitted by the vents serves as a vital nutrient source for symbiotic bacteria. These bacteria colonize the tissues of giant tube worms, which lack mouths and digestive tracts. Consequently, the tube worms are entirely dependent on these internal microbes for survival, as the bacteria convert the chemicals into organic molecules that nourish the worms.

According to the passage, the immediate precipitation of dissolved minerals on the ocean floor is caused by which of the following?

Passage

The evolution of the modern bicycle is a story of iterative mechanical improvements designed to increase speed, stability, and rider comfort. The journey began in 1817 when German baron Karl von Drais built a steerable, two-wheeled wooden vehicle. Patented in 1818 as the 'draisienne' (and often called the 'running machine' or 'dandy horse'), this device lacked pedals. Instead, riders sat astride a wooden frame and propelled themselves by pushing their feet against the ground in a walking motion. Although it demonstrated that a rider could balance on two wheels, the physical exertion required to operate it on rough carriage roads limited its widespread adoption.

For decades, the design remained a novelty until the early 1860s in Paris. Carriage makers Pierre and Ernest Michaux modified the design by attaching cranks and pedals directly to the front wheel's axle. This machine, called the 'velocipede,' allowed riders to propel themselves without touching the ground. However, because its frame was made of wrought iron and its wheels were wooden with iron bands, the vehicle transmitted every bump in the road directly to the rider. The resulting rough ride earned the velocipede the popular nickname of the 'boneshaker.'

In the 1870s, British engineers sought to make the bicycle faster. Because the pedals were still directly attached to the wheel axle, the only way to increase speed was to make the drive wheel larger. This led to the creation of the 'ordinary bicycle,' commonly known as the 'high-wheeler' or 'penny-farthing' because its wheels resembled the large penny and small farthing coins of the era. The front wheel grew to diameters of up to five feet, while the rear wheel shrank to a fraction of that size. These bicycles featured solid rubber tires and wire-spoked wheels, which slightly cushioned the ride. However, their high center of gravity made them extremely dangerous; hitting a stone could throw the rider forward over the handlebars in a crash known as a 'header.'

Recognizing the need for a safer design, English inventor John Kemp Starley introduced the Rover Safety Bicycle in 1885. The Rover featured two wheels of nearly equal size and a chain drive mechanism that connected the pedals to the rear wheel. This meant the bicycle could travel fast without requiring a dangerously large front wheel. The final major improvement came in 1888 when Scottish inventor John Boyd Dunlop patented the pneumatic (air-filled) rubber tire for bicycles. Dunlop’s invention replaced the jarring solid rubber tires, providing a smooth ride and securing the bicycle's place as a practical mode of daily transportation.

Based on the passage, in what chronological order did the following developments in bicycle design occur, from earliest to latest?

For decades, biologists studying the high-altitude forests of the Andes observed a mysterious decline in the local population of the golden-backed tree frog. Standard hypotheses pointed to shifting temperatures or increased solar radiation at high elevations. However, a 2018 study revealed a far more direct ecological chain reaction. The construction of a new highway through the valley below had introduced a non-native predatory beetle to the region. These beetles, which thrived in the cleared roadside brush, gradually migrated up the slopes. Because the golden-backed tree frog relies on a specific species of mountain bromeliad for shelter, the frogs were highly vulnerable. The predatory beetles did not eat the frogs; instead, they fed voraciously on the bromeliad leaves, destroying the plants. Deprived of their only nesting sites and protection from predators, the frogs suffered a rapid population collapse. This simple disruption of the local flora highlights how human infrastructure projects can have devastating downstream effects on specialized montane species.

Based on the passage, the population of the golden-backed tree frog collapsed because of which direct cause?

In triangle $PQR$, the measure of angle $P$ is $30^\circ$, the measure of angle $Q$ is $45^\circ$, and the length of side $QR$ is $10$ units. Which of the following expressions represents the length, in units, of side $PR$?