Let’s solve problem #10 first.
We are told that segments from the same external point tangent to a circle are congruent. That means if two lines touch the circle at one point each and come from the same outside point, they have the same length.
In problem #10:
- Point B is outside the circle.
- Segment BA touches the circle at A → so BA is tangent.
- Segment BC touches the circle at C → so BC is also tangent.
→ Therefore, BA ≅ BC (they are equal in length).
From the diagram:
- BA = 5x + 23
- BC = 8x - 19
Set them equal:
5x + 23 = 8x - 19
Now solve for x.
Subtract 5x from both sides:
23 = 3x - 19
Add 19 to both sides:
42 = 3x
Divide by 3:
x = 14
✔ Let’s check:
BA = 5(14) + 23 = 70 + 23 = 93
BC = 8(14) - 19 = 112 - 19 = 93
Yes, they match!
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Now let’s solve problem #11.
We’re asked to find PQ.
Looking at the diagram:
- Point Q is outside the circle.
- Segment QP touches the circle at P → tangent.
- Segment QR touches the circle at R → tangent.
→ So QP ≅ QR (same rule: tangents from same external point are congruent).
Given:
- QP = 14x - 13
- QR = 8x + 5
Set them equal:
14x - 13 = 8x + 5
Solve for x.
Subtract 8x from both sides:
6x - 13 = 5
Add 13 to both sides:
6x = 18
Divide by 6:
x = 3
Now plug back in to find PQ (which is QP):
QP = 14x - 13 = 14(3) - 13 = 42 - 13 = 29
Check with QR:
QR = 8x + 5 = 8(3) + 5 = 24 + 5 = 29 → matches!
So PQ = 29.
Final Answer:
29
Parent Tip: Review the logic above to help your child master the concept of tangent lines worksheet answers.
