Proofs About Angles Practice

Question

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Question
Use the image to answer the question.

The outline of a kite is in the shape of a diamond. The vertices of this diamond are labeled upper F, upper G, upper K, and upper J, respectively. Each of these points are connected via lines that intersect at a central, interior point: upper H.

In quadrilateral FGKJ
, FK¯¯¯¯¯¯¯¯
is a perpendicular bisector of GJ¯¯¯¯¯¯¯
. The length of segment GF¯¯¯¯¯¯¯¯
is 6+x
units, and the length of segment JF¯¯¯¯¯¯¯
is 3x−5
units. What is the value of x?

(1 point)
The value of x is
.

Answer 1

To find the value of \( x \), we begin by using the information provided about the segments in the kite. Since \( FK \) is the perpendicular bisector of \( GJ \), it splits \( GJ \) into two equal segments. This means that:

\[ GF = JF \]

We are given:

• The length of segment \( GF \) is \( 6 + x \).
• The length of segment \( JF \) is \( 3x - 5 \).

Setting these two expressions equal gives us:

\[ 6 + x = 3x - 5 \]

Now, we'll solve for \( x \). First, rearrange the equation:

\[ 6 + 5 = 3x - x \]

This simplifies to:

\[ 11 = 2x \]

Now, divide both sides by 2:

\[ x = \frac{11}{2} = 5.5 \]

Thus, the value of \( x \) is \( 5.5 \).