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Answer:
∆BCR ≅ ∆BAR
x = 6
Explanation:
For congruence, define point R where the altitude line meets segment AC.
The diagram shows you AR ≅ RC. You know BR ≅ BR by the reflexive property, and you know that ∠BRC ≅ ∠BRA = 90°. You can therefore claim ...
∆BCR ≅ ∆BAR . . . . by the SAS congruence postulate
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To solve for x, you don't need to show congruence of the right triangles. You only need to realize that the markings on segment AC show the left half is the same length (x) as the right half. The Pythagorean theorem is used to find x.
x² +8² = 10²
x² = 100 -64 = 36
x = √36 = 6
The length of x is 6 units.
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As you can see here, an altitude that is a median divides the isosceles triangle into two congruent right triangles. (That altitude also bisects angle B.)
You may recognize that the side ratios in the right triangle are 8:10 = 4:5. The missing side completes the Pythagorean triple 3:4:5, so is 2·3 = 6 = x.