Answer:
11) KI = 28
12) PR = 22
Explanation:
You want the base segment lengths KI and PR, given relations between those segments and the corresponding midsegment of the triangle.
Midsegment
A midsegment is a line segment that joins midpoints of other line segments, usually the midpoints of triangle sides or opposite sides of a quadrilateral. In the case of a triangle, the midsegment is parallel to the third side, and half its length. This relation is used to solve these problems.
11)
KI = 2·AB
(x +37) = 2(x +23) . . . . substitute given expressions
x +37 = 2x +46 . . . . . . eliminate parentheses
37 = x +46 . . . . . . . . . . subtract x
28 = x +37 . . . . . . . . . . subtract 9
The length of KI is 28 units.
12)
PR = 2·HG
(x +30) = 2(19 +x) . . . . substitute given expressions
x +30 = 38 +2x . . . . . eliminate parentheses
22 = x +30 . . . . . . . . subtract (x+8)
The length of PR is 22 units.
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Additional comment
As you can see here, there is no real reason to find the value of x, then substitute it back into the expression for the segment of interest. We can, and did, solve directly for the value of that expression.
As with any algebraic manipulation, any operation performed on one side of the equal sign must also be performed on the other side of the equal sign. When we say "subtract ()", we mean "subtract () from both sides of the equation."
In problem 11, x=-9. In problem 12, x=-8. Negative values of x work just fine, as long as the resulting segment lengths are positive.