Looking at the image, we are presented with a geometry problem from an “Arcs & Angles in Circles Scavenger Hunt.” The specific card shown is labeled
J and asks:
>
What is the measure of arc QTP?
The diagram shows:
- A circle.
- Two tangent lines drawn from an external point
R, touching the circle at points
Q and
T.
- The angle formed by the two tangents at point R is labeled
68°.
- We are to find the measure of the
major arc QTP — that is, the long arc going from Q to P to T (passing through P, which lies on the major arc).
---
Step-by-Step Solution:
#### 1. Understand the Geometry
When two tangents are drawn from an external point to a circle, they form an angle whose measure is related to the arcs intercepted.
Specifically, the
angle between two tangents from an external point is equal to
half the difference of the measures of the intercepted arcs.
In this case, the two tangents from point R intercept:
- The
minor arc QT (the short arc between Q and T)
- The
major arc QTP (the long arc from Q to T passing through P)
Let’s denote:
- Let \( x \) = measure of minor arc QT
- Then the major arc QTP = \( 360^\circ - x \)
The formula for the angle formed by two tangents is:
\[
\text{Angle} = \frac{1}{2} \left( \text{major arc} - \text{minor arc} \right)
\]
So,
\[
68^\circ = \frac{1}{2} \left( (360^\circ - x) - x \right)
\]
Simplify inside the parentheses:
\[
68^\circ = \frac{1}{2} (360^\circ - 2x)
\]
Multiply both sides by 2:
\[
136^\circ = 360^\circ - 2x
\]
Subtract 360° from both sides:
\[
-224^\circ = -2x
\]
Divide by -2:
\[
x = 112^\circ
\]
So, the
minor arc QT = 112°
Therefore, the
major arc QTP = 360° - 112° = 248°
---
✔ Final Answer: 248°
This is the measure of arc QTP.
---
Why This Works:
The key theorem used here is:
>
The measure of the angle formed by two tangents drawn from an external point is half the difference of the measures of the intercepted arcs.
Since the two tangents intercept both the minor and major arcs between their points of tangency, and the major arc is always larger, we subtract the minor arc from the major arc and take half — which gives us the angle at the external point.
We solved algebraically to find the minor arc first, then subtracted it from 360° to get the major arc QTP.
---
✔ So, the answer to card
J is:
248°
*(Note: The "Previous Answer: 209°" on card J is likely the answer to the previous card in the scavenger hunt — not relevant to solving this one.)*
Parent Tip: Review the logic above to help your child master the concept of circles and arcs worksheet answers.
