**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

Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit the following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**