What value of [tex]$x$[/tex] would make [tex]$\overline{ KM } / / \overline{ JN }$[/tex]? Complete the statements to solve for [tex]$x$[/tex]. By the converse of the side-splitter theorem, if JK/KL = [$\square$] then KM // JN. Substitute the expressions into the proportion: [tex]$\frac{x-5}{x}=\frac{x-3}{x+4}$[/tex]. Cross-multiply: [tex]$(x-5)$[/tex] ([$\square$] [tex]$=x(x-3)$[/tex]. Distribute: [tex]$x(x)+x(4)$[/tex] Multiply and simplify: [tex]$x^{ X }$[/tex] Solve for [tex]$x: x=$[/tex] [$\square$] [tex]$x+4$[/tex] [tex]$x-3$[/tex]
Answer (2)
By the converse of the side-splitter theorem, establish the proportion K L J K = J N A J , which gives x x − 5 = x + 4 x − 3 .
Cross-multiply to get ( x − 5 ) ( x + 4 ) = x ( x − 3 ) .
Distribute and simplify the equation to x 2 − x − 20 = x 2 − 3 x , which simplifies to 2 x = 20 .
Solve for x , obtaining x = 10 , thus the final answer is 10 .
Explanation
Analyze the problem and given data. Let's analyze the problem. We are given a geometric setup where we need to find the value of x that makes K M ∥ J N . The problem utilizes the converse of the side-splitter theorem, which states that if a line divides two sides of a triangle proportionally, then the line is parallel to the third side. We are given the proportion x x − 5 = x + 4 x − 3 , and we need to solve for x by completing the given statements.
Apply the converse of the side-splitter theorem. By the converse of the side-splitter theorem, if K L J K = J N A J , then K M ∥ J N . In our case, K L J K = x x − 5 and J N A J = x + 4 x − 3 . Therefore, the first blank should be J N A J which is equal to x − 3 .
Set up the proportion and identify the missing term. The proportion is given as x x − 5 = x + 4 x − 3 . We need to cross-multiply to solve for x . Cross-multiplication gives us ( x − 5 ) ( x + 4 ) = x ( x − 3 ) . Therefore, the second blank should be x + 4 .
Distribute the terms. Now, let's distribute the terms on both sides of the equation:
On the left side: ( x − 5 ) ( x + 4 ) = x ( x ) + x ( 4 ) − 5 ( x ) − 5 ( 4 ) = x 2 + 4 x − 5 x − 20 = x 2 − x − 20 .
On the right side: x ( x − 3 ) = x ( x ) − 3 x = x 2 − 3 x .
So, we have x 2 − x − 20 = x 2 − 3 x .
Solve for x. Now, let's simplify the equation and solve for x :
x 2 − x − 20 = x 2 − 3 x
Subtract x 2 from both sides: − x − 20 = − 3 x
Add 3 x to both sides: 2 x − 20 = 0
Add 20 to both sides: 2 x = 20
Divide by 2 : x = 10
Determine the power of x. The question asks to find x X after distributing x ( x ) + x ( 4 ) . Since x ( x ) + x ( 4 ) = x 2 + 4 x , the highest power of x is 2. Therefore, X = 2 .
State the final answer. The value of x that makes K M ∥ J N is x = 10 .
Examples
In architecture, the side-splitter theorem and its converse are used to divide spaces proportionally. For example, when designing a staircase, the theorem helps ensure that the steps are evenly spaced and proportional, creating a visually appealing and structurally sound design. By applying the principles of proportional division, architects can create harmonious and balanced spaces that enhance the overall aesthetic and functionality of a building.
The value of x that makes the segments K M parallel to J N is x = 10 . This is found using the converse of the side-splitter theorem and solving the equation derived from the proportion. The highest power of x in the equation after distribution is 2.
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