Geometry: Using the Pythagorean Theorem in Real-World Problems (CCSS.8.G.7)
Common Core 8th Grade Math · Learn by Concept
Help Questions
Common Core 8th Grade Math › Geometry: Using the Pythagorean Theorem in Real-World Problems (CCSS.8.G.7)
A right triangle has legs 9 cm and 12 cm. What is the length of the missing side (the hypotenuse)?
21 cm
15 cm
225 cm
12.4 cm
Explanation
Use the Pythagorean Theorem $a^2 + b^2 = c^2$. Here $c$ is the hypotenuse: $c^2 = 9^2 + 12^2 = 81 + 144 = 225$, so $c = \sqrt{225} = 15$ cm. Distractors: adding legs (21), forgetting the square root (225), or mis-squaring then taking $\sqrt{153} \approx 12.4$.
A 10-foot ramp reaches a platform 6 feet high, forming a right triangle with the ground. What is the horizontal distance from the platform to the end of the ramp?
8 ft
64 ft
11.7 ft
9.4 ft
Explanation
Let the horizontal distance be $x$. Using $a^2 + b^2 = c^2$ with hypotenuse $c=10$ and vertical leg $6$: $x^2 + 6^2 = 10^2 \Rightarrow x^2 = 100 - 36 = 64$, so $x = \sqrt{64} = 8$ ft. Distractors: forgetting the square root (64), adding instead of subtracting $\sqrt{100+36} \approx 11.7$, or mis-squaring $6^2$ as $12$ giving $\sqrt{88} \approx 9.4$.
A kite string is 25 meters long and is taut, forming a right triangle with the ground. If the horizontal distance from the person to the point below the kite is 24 meters, what is the height of the kite?
34.6 m
49 m
7 m
19.6 m
Explanation
Let the height be $h$. With hypotenuse $25$ and base $24$: $h^2 + 24^2 = 25^2 \Rightarrow h^2 = 625 - 576 = 49$, so $h = \sqrt{49} = 7$ m. Distractors: adding instead of subtracting $\sqrt{625+576} \approx 34.6$, forgetting the square root (49), or mis-squaring $24^2$ as $240$ giving $\sqrt{385} \approx 19.6$.
A 15-foot ladder leans against a wall. The foot of the ladder is 9 feet from the base of the wall. What is the height up the wall that the ladder reaches?
17
14
144
12
Explanation
The ladder and wall form a right triangle with hypotenuse 15 and base 9. Height $=\sqrt{15^2-9^2}=\sqrt{225-81}=\sqrt{144}=12$. Distractors: adding instead of subtracting inside the square root (gives a number near 17), mis-squaring a leg (yields a height near 14), or forgetting the square root (reporting $144$).
A 13-foot ladder rests against a vertical wall, with its base 5 feet from the wall. The ladder, the wall, and the ground form a right triangle. What is the length of the missing vertical side (how high up the wall does the ladder reach)?
18 ft
13.9 ft
12 ft
144 ft
Explanation
Let the height be $h$. By the Pythagorean Theorem, $h^2 + 5^2 = 13^2$. So $h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$ ft.
In a right triangle, the hypotenuse is 10 units and one leg is 6 units. What is the length of the missing leg?
8
64
4
11.7
Explanation
Let the missing leg be $x$. Then $x^2 + 6^2 = 10^2$. So $x = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$.
A rectangular park is 24 meters by 7 meters. A straight path is planned from one corner to the opposite corner. The path forms the hypotenuse of a right triangle. What is the length of the missing side (the diagonal path)?
31 m
22.9 m
625 m
25 m
Explanation
The diagonal $d$ of a rectangle forms a right triangle with legs 24 and 7. So $d = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25$ m.
A rectangular box measures 6 inches by 8 inches by 24 inches. The segment from one bottom corner to the opposite top corner is the space diagonal and forms right triangles in three dimensions. What is the length of this missing space diagonal?
38 in
26 in
11.1 in
676 in
Explanation
Use the 3D Pythagorean Theorem: $d = \sqrt{6^2 + 8^2 + 24^2} = \sqrt{36 + 64 + 576} = \sqrt{676} = 26$ in.
In a right triangle, the legs are 9 and 12. What is the length of the missing side (the hypotenuse)?
21
13
15
225
Explanation
Use $a^2+b^2=c^2$. So $c=\sqrt{9^2+12^2}=\sqrt{81+144}=\sqrt{225}=15$. (21 adds the legs, 225 forgets the square root, 13 confuses it with a $5$-$12$-$13$ triangle.)
In a right triangle, the hypotenuse is 13 and one leg is 5. What is the length of the other leg?
12
18
$\sqrt{194}$
144
Explanation
Let the unknown leg be $b$. By $a^2+b^2=c^2$, $b=\sqrt{13^2-5^2}=\sqrt{169-25}=\sqrt{144}=12$. (18 adds lengths, $\sqrt{194}$ adds squares instead of subtracting, 144 forgets the square root.)