A right triangle has legs with lengths of $10, 2 \sqrt{35}$, and $2 \sqrt{d}$. If $d<35$, what is the value of $d$?
Answer (2)
We analyze the problem using the Pythagorean theorem a 2 + b 2 = c 2 , considering three cases for which sides are legs and which is the hypotenuse.
In Case 1, where 10 and 2 35 are legs, we find d = 60 , which contradicts d < 35 .
In Case 2, where 10 and 2 d are legs, we find d = 10 , which satisfies d < 35 .
In Case 3, where 2 35 and 2 d are legs, we find d = − 10 , which is not possible since side lengths must be positive. Therefore, the final answer is 10 .
Explanation
Problem Analysis and Setup We are given a right triangle with side lengths 10 , $2
\sqrt{35}$, and $2
\sqrt{d} , w h ere d < 35 . W e n ee d t o f in d t h e v a l u eo f d . I na r i g h tt r ian g l e , t h e P y t ha g ore an t h eore m s t a t es t ha t a^2 + b^2 = c^2 , w h ere a an d b a re t h e l e n g t h so f t h e l e g s an d c i s t h e l e n g t h o f t h e h y p o t e n u se . W e w i ll co n s i d er t h ree p oss ib l ec a ses t o d e t er min e t h e v a l u eo f d$.
Case 1: 10 and 2sqrt(35) are legs Case 1: 10 and $2
\sqrt{35}$ are the legs, and $2
\sqrt{d}$ is the hypotenuse.
Then, we have:
1 0 2 + ( 2 35 ) 2 = ( 2 d ) 2
100 + 4 ( 35 ) = 4 d
100 + 140 = 4 d
240 = 4 d
d = 60
Since we are given that d < 35 , this case is not possible.
Case 2: 10 and 2sqrt(d) are legs Case 2: 10 and $2
\sqrt{d}$ are the legs, and $2
\sqrt{35}$ is the hypotenuse.
Then, we have:
1 0 2 + ( 2 d ) 2 = ( 2 35 ) 2
100 + 4 d = 4 ( 35 )
100 + 4 d = 140
4 d = 40
d = 10
Since d = 10 < 35 , this case is possible.
Case 3: 2sqrt(35) and 2sqrt(d) are legs Case 3: $2
\sqrt{35}$ and $2
\sqrt{d}$ are the legs, and 10 is the hypotenuse.
Then, we have:
( 2 35 ) 2 + ( 2 d ) 2 = 1 0 2
4 ( 35 ) + 4 d = 100
140 + 4 d = 100
4 d = − 40
d = − 10
Since d must be positive (because we have $2
\sqrt{d}$ as a side length), this case is not possible.
Conclusion Therefore, the only possible value for d is 10 .
Examples
The Pythagorean theorem is a fundamental concept in construction and architecture. For example, when building a house, ensuring that the walls are perfectly perpendicular to the foundation is crucial for structural integrity. Builders use the 3-4-5 right triangle rule (a multiple of which is 6-8-10) to verify square corners, ensuring that the foundation and walls form a precise 90-degree angle. This guarantees stability and prevents future structural problems.
The value of d in the right triangle configuration is 10 . This was determined using the Pythagorean theorem by analyzing three different cases for the triangle's sides. Only one case resulted in a valid value that meets the condition d < 35 .
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