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Pythagorean Theorem Word Problems

7 soru

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Soru 1Soru →

A rescue helicopter flies 12 miles due south and then 35 miles due west to reach a medical emergency. After the pickup, the helicopter flies directly back to its base in a straight line. What is the total distance the helicopter traveled during this entire trip?

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Cevap: 84 miles

Cevap

84 miles
The total distance is the sum of the outbound journey and the return journey. The outbound journey consists of the two legs of the flight: 12 + 35 = 47 miles. The return journey is a straight line that forms the hypotenuse of a right triangle. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), the return distance is 122+352=1369=37\sqrt{12^2 + 35^2} = \sqrt{1369} = 37 miles. The total distance traveled is 47 + 37 = 84 miles.

Adım Adım Çözüm

1
Calculate the distance traveled during the outbound trip.
12+35=4712 + 35 = 47 miles
The helicopter travels along two perpendicular paths (south and west) to reach the destination.
2
Set up the Pythagorean theorem to find the straight-line return distance.
c2=122+352c^2 = 12^2 + 35^2
The south and west paths form a right angle, making the direct return path the hypotenuse of a right triangle.
3
Solve for the return distance (the hypotenuse).
c=144+1225=1369=37c = \sqrt{144 + 1225} = \sqrt{1369} = 37 miles
Taking the square root of the sum of the squares gives the length of the straight-line return trip.
4
Add the outbound distance and the return distance.
47+37=8447 + 37 = 84 miles
The question asks for the total distance traveled during the entire round trip.

Anahtar Kavram

Applying the Pythagorean theorem to find a missing distance in a multi-step navigation problem.
Soru 2Soru →

A wooden ramp used for moving equipment is 17 feet long. If the horizontal distance from the base of the ramp to the building is 15 feet, what is the vertical height of the top of the ramp?

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Cevap: 8 feet

Cevap

8 feet
The scenario describes a right triangle where the ramp is the hypotenuse, and the distance to the building is one of the legs. Using the Pythagorean theorem, a2+b2=c2a^2 + b^2 = c^2. Plugging in the knowns gives 152+b2=17215^2 + b^2 = 17^2, which evaluates to 225+b2=289225 + b^2 = 289. Subtracting 225 from both sides yields b2=64b^2 = 64. The square root of 64 is 8, so the ramp reaches a vertical height of 8 feet.

Adım Adım Çözüm

1
Identify the known and unknown sides of the right triangle formed by the ramp, the ground, and the building.
The hypotenuse (ramp) is c=17c = 17, the horizontal leg (ground) is a=15a = 15, and the vertical leg (height) is bb.
To determine which part of the Pythagorean theorem formula (a2+b2=c2a^2 + b^2 = c^2) needs to be solved.
2
Substitute the known values into the Pythagorean theorem and square them.
152+b2=17215^2 + b^2 = 17^2 becomes 225+b2=289225 + b^2 = 289.
To set up the equation for isolating the unknown variable.
3
Subtract 225 from 289 to isolate b2b^2, then take the square root.
b2=289−225=64b^2 = 289 - 225 = 64. Taking the square root gives b=64=8b = \sqrt{64} = 8.
To find the final linear measurement of the vertical height.

Anahtar Kavram

Applying the Pythagorean theorem to find a missing leg of a right triangle.
Tahmini Süre:45s
Soru 3Soru →

A builder uses a 15-foot diagonal support beam to brace a vertical wall. If the base of the beam rests on the flat floor exactly 9 feet away from the bottom of the wall, at what height on the wall, in feet, does the top of the beam attach?

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Cevap: 12

Cevap

12
The physical scenario forms a right triangle where the hypotenuse (the beam) measures 15 feet and one leg (along the floor) measures 9 feet. By using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we can find the missing leg by solving 92+b2=1529^2 + b^2 = 15^2. This simplifies to 81+b2=22581 + b^2 = 225, which means b2=144b^2 = 144. Taking the square root of 144 gives a wall height of 12 feet.

Adım Adım Çözüm

1
Identify the parts of the right triangle.
Hypotenuse c=15c = 15 feet, base leg a=9a = 9 feet, vertical leg bb is unknown.
The beam forms the hypotenuse opposite the right angle made by the intersection of the vertical wall and flat floor.
2
Apply the Pythagorean theorem.
92+b2=1529^2 + b^2 = 15^2
The Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) relates the lengths of the legs of a right triangle to its hypotenuse.
3
Square the known values.
81+b2=22581 + b^2 = 225
To solve for the unknown leg bb, we must first evaluate the squared terms.
4
Isolate the unknown squared variable.
b2=225−81=144b^2 = 225 - 81 = 144
Subtracting 81 from both sides leaves b2b^2 by itself.
5
Take the square root to find the height.
b=144=12b = \sqrt{144} = 12
Taking the square root of b2b^2 gives the actual length of the wall height.

Anahtar Kavram

Using the Pythagorean theorem to find the missing leg of a right triangle.
Soru 4Soru →

A telecommunications crew is installing a straight fiber-optic cable from the base of a network tower to a control panel mounted on a nearby building. The horizontal distance from the tower base to the building is 1212 yards. The control panel is mounted 55 yards above the ground. How many feet long must the cable be to reach exactly from the tower base to the control panel?

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Cevap: 39

Cevap

39
The scenario creates a right triangle with legs measuring 55 yards and 1212 yards. Using the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we find the hypotenuse is 52+122=25+144=169=13\sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 yards. Because the question asks for the answer in feet, we multiply 1313 yards by 33 feet per yard to get exactly 3939 feet.

Adım Adım Çözüm

1
Identify the shape formed by the scenario.
The ground, the building wall, and the cable form a right triangle.
This allows the use of the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find the length of the diagonal cable.
2
Calculate the length of the cable in yards.
52+122=25+144=1695^2 + 12^2 = 25 + 144 = 169. The square root of 169169 is 1313 yards.
Applying the Pythagorean theorem using the given leg lengths of 55 yards and 1212 yards yields the hypotenuse.
3
Convert the final length from yards to feet.
13 yards×3 feet/yard=39 feet13 \text{ yards} \times 3 \text{ feet/yard} = 39 \text{ feet}.
The question specifically asks for the length of the cable in feet.

Anahtar Kavram

Pythagorean Theorem and Unit Conversions
Tahmini Süre:1m 0s
Soru 5Soru →

A military training field is rectangular, measuring 160160 yards wide and 300300 yards long. A recruit must travel from one corner of the field to the opposite corner. If the recruit runs directly across the diagonal instead of moving along the outer edges of the two sides, how many yards does the recruit save?

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Cevap: 120

Cevap

120 yards
The route along the edges is the sum of the width and length (160+300=460160 + 300 = 460 yards). The diagonal route is the hypotenuse of the right triangle formed by these sides, calculated as 1602+3002=340\sqrt{160^2 + 300^2} = 340 yards. The distance saved is 460−340=120460 - 340 = 120 yards.

Adım Adım Çözüm

1
Calculate the total distance if traveling along the two outer edges of the rectangular field.
160+300=460160 + 300 = 460 yards.
This establishes the longer route distance to compare against the diagonal shortcut.
2
Set up the Pythagorean theorem to find the length of the diagonal route.
1602+3002=c2160^2 + 300^2 = c^2
The rectangular field forms a right triangle where the width and length are the legs, and the diagonal is the hypotenuse.
3
Solve for the diagonal length (cc).
c=115600=340c = \sqrt{115600} = 340 yards.
This gives the exact distance of the direct route.
4
Calculate the difference between the two routes.
460−340=120460 - 340 = 120 yards.
Subtracting the shorter diagonal route from the longer perimeter route yields the exact distance saved.

Anahtar Kavram

Applying the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find a direct diagonal distance, and comparing it to the linear distance along the edges.
Soru 6Soru →

A surveyor is mapping a plot of land and launches a drone from a perfectly flat field. The drone ascends vertically to an altitude of 3030 meters. It then travels horizontally in a straight line for 1616 meters to photograph a landmark. What is the direct line-of-sight distance, in meters, from the launch point on the ground to the drone?

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Cevap: 34

Cevap

The direct line-of-sight distance is 3434 meters.
The drone's flight path forms a right-angled triangle relative to the starting point. By applying the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2), we square the vertical distance (302=90030^2 = 900) and the horizontal distance (162=25616^2 = 256). Adding these values yields 11561156. Taking the square root of 11561156 gives the direct line-of-sight distance, which is 3434 meters.

Adım Adım Çözüm

1
Model the drone's flight path as a right-angled triangle.
The vertical ascent forms one leg (a=30a = 30) and the horizontal flight forms the other leg (b=16b = 16). The direct line-of-sight distance is the hypotenuse (cc).
Vertical and horizontal movements are perpendicular, creating a 90∘90^\circ angle.
2
Apply the Pythagorean theorem to calculate the hypotenuse.
a2+b2=c2a^2 + b^2 = c^2 becomes 302+162=c230^2 + 16^2 = c^2.
The Pythagorean theorem relates the lengths of the legs to the length of the hypotenuse in any right triangle.
3
Square the lengths of the two legs and add them together.
900+256=1156900 + 256 = 1156. So, c2=1156c^2 = 1156.
This evaluates the left side of the equation to isolate c2c^2.
4
Take the square root of the sum to find the actual distance.
c=1156=34c = \sqrt{1156} = 34.
Taking the square root of c2c^2 yields the length of the hypotenuse.

Anahtar Kavram

Applying the Pythagorean theorem to find the hypotenuse in a practical right-triangle scenario.
Soru 7Soru →

A zip line is being installed from an observation deck down to a ground anchor. The observation deck is 4545 yards directly above the base of the structure. The ground anchor is secured on flat ground 2424 yards horizontally away from the base. Assuming the zip line is pulled perfectly straight, what is the required length of the zip line cable in feet?

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Cevap: 153

Cevap

153
The correct answer accurately calculates the diagonal length using the Pythagorean theorem (452+242=c245^2 + 24^2 = c^2, so c=51c = 51 yards) and then converts that distance into feet by multiplying by 3 (51×3=15351 \times 3 = 153 feet).

Adım Adım Çözüm

1
Apply the Pythagorean theorem (a2+b2=c2a^2 + b^2 = c^2) to find the diagonal length of the zip line in yards.
242+452=576+2025=260124^2 + 45^2 = 576 + 2025 = 2601 square yards.
The vertical height and horizontal distance form a right triangle with the zip line acting as the hypotenuse.
2
Calculate the square root of the sum to find the actual length.
2601=51\sqrt{2601} = 51 yards.
This isolates cc (the hypotenuse) to give the actual length of the zip line in yards.
3
Convert the length from yards to feet.
51×3=15351 \times 3 = 153 feet.
The question specifically asks for the length in feet, and there are 33 feet in one yard.

Anahtar Kavram

Using the Pythagorean theorem to find a hypotenuse and converting units of measurement.
Pythagorean Theorem Word Problems Alıştırma Soruları — Armed Services Vocational Aptitude Battery (ASVAB) | Examkin