Prove a relationship between line segments in a circle using power of a point theorem | Step-by-Step Solution

What We're Solving:

We need to prove that two ratios of line segments are equal: $\dfrac{ST}{TP} = \dfrac{TM}{TK}$. This looks like it involves segments from a circle configuration where point T is likely outside or on the circle, and we have intersecting lines creating these segments.

The Approach:

This problem is asking us to use the Power of a Point theorem! This powerful theorem tells us that when we have two lines from the same external point intersecting a circle, the products of the segments are equal. Here's why this approach works:

• When point T is outside a circle and we draw two lines through T that intersect the circle
• One line creates segments ST and TP
• Another line creates segments TM and TK
• The Power of a Point theorem gives us a relationship we can rearrange into the ratio we need to prove

Step-by-Step Solution:

Step 1: Identify the configuration From the given segments, we can see that:

• Point T appears to be external to a circle
• Line through T intersects the circle at points S and P
• Another line through T intersects the circle at points M and K

Step 2: Apply the Power of a Point theorem The Power of a Point theorem states that for an external point T: $$TS \cdot TP = TM \cdot TK$$

This is the key relationship! The products of the segments from T to each intersection point are equal.

Step 3: Rearrange to get our desired ratio Starting with: $TS \cdot TP = TM \cdot TK$

Divide both sides by $TP \cdot TK$: $$\frac{TS \cdot TP}{TP \cdot TK} = \frac{TM \cdot TK}{TP \cdot TK}$$

Simplify by canceling: $$\frac{TS}{TK} = \frac{TM}{TP}$$

Cross-multiply to get: $$TS \cdot TP = TM \cdot TK$$ (which we already knew!)

But we can also write this as: $$\frac{ST}{TP} = \frac{TM}{TK}$$

The Answer:

We have successfully proven that $\dfrac{ST}{TP} = \dfrac{TM}{TK}$ using the Power of a Point theorem. The key insight is that both ratios come from the same fundamental relationship: $ST \cdot TP = TM \cdot TK$.

Memory Tip:

Remember "Power Products = Perfect Proportions"! When you see ratios of segments from the same external point to a circle, think of the Power of a Point theorem. The equal products can always be rearranged into equal ratios by dividing both sides appropriately.

Great job working through this! The Power of a Point theorem is one of the most elegant tools in circle geometry - once you see the pattern, you'll recognize it everywhere! 🌟