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What is the value of x?
5 units
2 units
3 units
8 units
(Image: Triangle A B C is shown. Angle A B C is a right angle. An altitude is drawn from point B to point D on side A C to form a right angle. The length of A D is x, the length of D C is 4 x, and the length of B D is 10.)
To find the value of ( x ) in the triangle formed by points ( A, B, C, ) and ( D ), we can use the information given about the lengths.
1. The total length of side ( AC ) is the sum of lengths ( AD ) and ( DC ):
[
AC = AD + DC = x + 4x = 5x
]
2. According to the problem, ( BD ) is the height from point ( B ) to side ( AC ), and its length is given as 10.
3. Now we can use the Pythagorean theorem in the triangles ( ABD ) and ( BDC ):
– For triangle ( ABD ):
[
AB^2 = AD^2 + BD^2 Rightarrow AB^2 = x^2 + 10^2 = x^2 + 100
]
– For triangle ( BDC ):
[
BC^2 = DC^2 + BD^2 Rightarrow BC^2 = (4x)^2 + 10^2 = 16x^2 + 100
]
4. Since ( AB ) and ( BC ) are the same sides (it’s one triangle), we can set the equations equal to each other:
[
x^2 + 100 = 16x^2 + 100