Question

su and vt are chords that intersect at point r. what is the length of line segment vt? 4 units 8 units 14 units 13 units

Explanation:

Step1: Apply the intersecting - chords theorem

The intersecting - chords theorem states that if two chords \(SU\) and \(VT\) intersect at a point \(R\) inside a circle, then \(VR\times RT=SR\times RU\). Here, \(VR = x + 1\), \(RT=x + 4\), \(SR=x + 6\), and \(RU=x\). So, \((x + 1)(x + 4)=(x + 6)x\).

Step2: Expand both sides of the equation

Expand the left - hand side: \((x + 1)(x + 4)=x^{2}+4x+x + 4=x^{2}+5x + 4\). Expand the right - hand side: \((x + 6)x=x^{2}+6x\).

Step3: Solve the resulting equation

Set the two expanded expressions equal to each other: \(x^{2}+5x + 4=x^{2}+6x\). Subtract \(x^{2}\) from both sides of the equation: \(5x + 4=6x\). Then, subtract \(5x\) from both sides to get \(x = 4\).

Step4: Find the length of \(VT\)

Since \(VT=VR+RT\), and \(VR=x + 1\), \(RT=x + 4\), substitute \(x = 4\) into these expressions. \(VR=4 + 1=5\) and \(RT=4 + 4=8\). So, \(VT=5 + 8=13\) units.

Answer:

13 units