**Bisectors of a Triangle Worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on bisectors of a triangle.

**Problem 1 :**

Construct a perpendicular bisector to a line segment.

**Problem 2 :**

Construct the circumcenter of the triangle ABC with AB = 5 cm, **∠**A = 70° and **∠**B = 70°.

**Problem 3 :**

A company plans to build a distribution center that is convenient to three of its major clients as shown in the diagram below.

The planners start by roughly locating the three clients on a sketch and finding the circumcenter of the triangle formed.

(i) Explain why using the circumcenter as the location of a distribution center would be convenient for all the clients.

(ii) Make a sketch of the triangle formed by the clients. Locate the circumcenter of the triangle. Tell what segments are congruent.

**Problem 4 :**

In the diagram shown below, the angle bisectors of ΔMNP meet at point L.

(i) What segments are congruent ?

(ii) Find LQ and LR

**Problem 1 :**

Construct a perpendicular bisector to a line segment.

**Solution : **

**Step 1 :**

Draw the line segment AB.

**Step 2 :**

With the two end points A and B of the line segment as centers and more than half the length of the line segment as radius draw arcs to intersect on both sides of the line segment at C and D.

**Step 3 :**

Join C and D to get the perpendicular bisector of the given line segment AB.

In the diagram above, CD is the perpendicular bisector of the line segment AB.

**Problem 2 :**

Construct the circumcenter of the triangle ABC with AB = 5 cm, **∠**A = 70° and **∠**B = 70°.

**Solution : **

**Step 1 :**

Draw triangle ABC with the given measurements.

**Step 2 :**

Construct the perpendicular bisectors of any two sides (AC and BC) and let them meet at S which is the circumcentre.

**Problem 3 :**

A company plans to build a distribution center that is convenient to three of its major clients as shown in the diagram below.

The planners start by roughly locating the three clients on a sketch and finding the circumcenter of the triangle formed.

(i) Explain why using the circumcenter as the location of a distribution center would be convenient for all the clients.

(ii) Make a sketch of the triangle formed by the clients. Locate the circumcenter of the triangle. Tell what segments are congruent.

**Solution (i) :**

Because the circumcenter is equidistant from the three vertices, each client would be equally close to the distribution center.

**Solution (ii) :**

Label the vertices of the triangle as E, F, and G. Draw the perpendicular bisectors. Label their intersection as D.

By theorem 1 given above, in a triangle, the perpendicular bisectors intersect at a point that is equidistant from the vertices of the triangle.

So,

DE = DF = DG

**Problem 4 :**

In the diagram shown below, the angle bisectors of ΔMNP meet at point L.

(i) What segments are congruent ?

(ii) Find LQ and LR

**Solution (i) : **

By theorem "Concurrency of Angle Bisectors of a Triangle", the three angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

So, we have

LR ≅ LQ ≅ LS

**Solution (ii) : **

By theorem "Concurrency of Angle Bisectors of a Triangle", the three angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle.

Use the Pythagorean Theorem to find LQ in ΔLQM.

(LQ)^{2} + (MQ)^{2} = (LM)^{2}

Substitute MQ = 15 and LM = 17.

(LQ)^{2} + (15)^{2} = (17)^{2}

Simplify.

(LQ)^{2} + 225 = 289

Subtract 225 from both sides.

(LQ)^{2} = 64

(LQ)^{2} = 8^{2}

LQ = 8 units

Because LR ≅ LQ,

LR = 8 units

After having gone through the stuff given above, we hope that the students would have understood "Bisectors of a triangle worksheet"

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