**Segment Lengths in Circles Worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on segment lengths in circles.

Before look at the worksheet, if you would like to know the stuff related to segment lengths in circles,

**Problem 1 :**

In the diagram shown below, prove the following.

EA **⋅ **EB = EC **⋅** ED

**Problem 2 :**

Chords ST and PQ intersect inside the circle. Find the value of x.

**Problem 3 : **

Find the value of x in the diagram shown below.

**Problem 4 :**

Find the value of x in the diagram shown below.

**Problem 5 :**

You are standing at point C, about 8 feet from a circular aquarium tank. The distance from you to a point of tangency on the tank is about 20 feet. Estimate the radius of the tank.

**Problem 1 :**

In the diagram shown below, prove the following.

EA **⋅ **EB = EC **⋅** ED

**Solution : **

We can use similar triangles to prove the Theorem.

Given : AB, CD are chords that intersect at E.

To Prove : EA · EB = EC · ED

Draw DB and AC in the above diagram.

Because m∠C and m∠B intercept the same are ∠C ≅ ∠B. Likewise ∠A ≅ ∠D.

By the AA Similarity Postulate. ∆AEC ∼ ∆DEB.

So, the lengths of corresponding sides are proportional.

EA/ED = EC/EB

EA **⋅ **EB = EC **⋅** ED

**Problem 2 :**

Chords ST and PQ intersect inside the circle. Find the value of x.

**Solution : **

Using Theorem, we have

RQ · RP = RS · RT

Substitute.

9 · x = 3 · 6

9x = 18

Divide each side by 9.

9x/9 = 18/9

x = 2

**Problem 3 : **

Find the value of x in the diagram shown below.

**Solution :**

Using Theorem, we have

RP · RQ = RS · RT

Substitute.

9 · (11 + 9) = 10 · (x + 10)

Simplify.

180 = 10x + 100

Subtract 100 from each side.

80 = 10x

Divide each side by 10.

80/10 = 10x/10

8 = x

**Problem 4 :**

Find the value of x in the diagram shown below.

**Solution :**

Using Theorem, we have

(BA)^{2} = BC · BD

Substitute.

6^{2} = x · (x + 5)

Simplify.

36 = x^{2} + 5x

Subtract 36 from each side.

0 = x^{2} + 5x - 36

or

x^{2} + 5x - 36 = 0

Factor.

(x + 9)(x - 4) = 0

x + 9 = 0 or x - 4 = 0

x = - 9 or x = 4

We can use only positive value for x. because lengths cannot be negative.

So, we have

x = 4

**Problem 5 :**

You are standing at point C, about 8 feet from a circular aquarium tank. The distance from you to a point of tangency on the tank is about 20 feet. Estimate the radius of the tank.

**Solution :**

Using Theorem, we have

(CB)^{2} = CE · CD

Substitute.

20^{2} ≈ 8 · (2r + 8)

Simplify.

400 ≈ 16r + 64

Subtract 64 from each side.

336 ≈ 16r

Divide each side by 16.

336/16 ≈ 16r/16

21 ≈ r

Hence, the radius of the tank is about 21 feet.

After having gone through the stuff given above, we hope that the students would have understood, "Segment Lengths in Circles Worksheet".

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