Activity 6: The intersecting segments of chords Power Theorem. Solve for x: 1) 12/13 = x/10, 2) 14/10 = x/8, 3) x/6 = 7/14, 4) 9/16 = x/10, 5) 10/x = 24/10, 6) x/20 = 20/10.

Steps to Solve

1. Understand the Power of a Point Theorem The Power of a Point Theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal. This can be expressed as: $$ (a \cdot b) = (c \cdot d) $$ where ( a ) and ( b ) are the segments of one chord, and ( c ) and ( d ) are the segments of the other chord.

2. Identify the variables for each problem

• For each of the 6 problems, identify the segments and set the equation based on the lengths provided.
• Example for the first problem: ( 10 \cdot x = 12 \cdot 13 ).

3. Solve the equations For each chord intersection, rearrange the equation to solve for ( x ).

• Continuing the first problem: $$ 10x = 12 \cdot 13 $$ Now, calculate ( 12 \cdot 13 ): $$ 12 \cdot 13 = 156 $$ So, the equation becomes: $$ 10x = 156 $$

4. Isolate ( x ) Divide both sides by 10 to find ( x ): $$ x = \frac{156}{10} $$ This simplifies to: $$ x = 15.6 $$

5. Repeat for all problems Continue solving each problem using the same method of setting up equations and isolating ( x ) based on the segment lengths given.