solve for ( x ). leave your answer in simplest radical form.

Explanation:

Step1: Find the height of the right triangle (the common leg)

Let the height of the right triangle (the leg common to both right triangles) be ( h ). For the smaller right triangle with legs ( 2 ) and ( 5 ), by the Pythagorean theorem, ( h^{2}+2^{2}=5^{2} )? Wait, no, the smaller right triangle has legs ( 2 ) and ( h ), and hypotenuse ( 5 )? Wait, no, looking at the diagram, the smaller right triangle has legs ( 2 ) and ( h ), and hypotenuse ( 5 )? Wait, no, the right angle is between ( 2 ) and ( h ), so ( h^{2}+2^{2}=5^{2} )? Wait, no, ( 2 ) and ( 5 ) are legs? Wait, the smaller right triangle has legs ( 2 ) and ( h ), and hypotenuse ( 5 )? Wait, no, the right angle is marked, so the legs are ( 2 ) and ( h ), hypotenuse ( 5 ). So ( h^{2}+2^{2}=5^{2} )? Wait, ( 2^{2}+h^{2}=5^{2} ), so ( h^{2}=25 - 4=21 ), so ( h=\sqrt{21} ). Wait, no, maybe the smaller right triangle has legs ( 2 ) and ( 5 )? No, the right angle is between ( 2 ) and the vertical leg, so the vertical leg is ( h ), horizontal leg ( 2 ), hypotenuse ( 5 ). So ( h^{2}=5^{2}-2^{2}=25 - 4 = 21 ), so ( h=\sqrt{21} ).

Step2: Use Pythagorean theorem for the larger right triangle

The larger right triangle has legs ( 7 ) and ( h=\sqrt{21} ), and hypotenuse ( x ). So by Pythagorean theorem, ( x^{2}=7^{2}+h^{2} ). Substitute ( h^{2}=21 ), so ( x^{2}=49 + 21=70 ), so ( x=\sqrt{70} ). Wait, wait, no, maybe I mixed up. Wait, the larger right triangle has leg ( 7 ) and leg ( h ), hypotenuse ( x ). Wait, the vertical leg is ( h ), horizontal leg ( 7 ), so ( x^{2}=7^{2}+h^{2} ). And ( h ) is the leg of the smaller right triangle with legs ( 2 ) and ( h ), hypotenuse ( 5 )? Wait, no, maybe the smaller right triangle has legs ( 2 ) and ( 5 )? No, the right angle is between ( 2 ) and the vertical leg, so vertical leg ( h ), horizontal leg ( 2 ), hypotenuse ( 5 ). So ( h^{2}=5^{2}-2^{2}=25 - 4 = 21 ). Then the larger right triangle has legs ( 7 ) and ( h ), so ( x^{2}=7^{2}+h^{2}=49 + 21 = 70 ), so ( x=\sqrt{70} ).

Answer:

( \sqrt{70} )