# Exercise E7.5 Triangles NCERT Solutions Class 9

## Question 1

\( ABC\) is a triangle. Locate a point in the interior of \(ΔABC\) which is equidistant from all the vertices of \(ΔABC.\)

### Solution

**Video Solution**

**Steps:**

Circumcentre of a triangle is always equidistant from all the vertices of that triangle.

Circumcentre is the point where perpendicular bisectors of all the sides of the triangle meet together.

In \(\Delta ABC\), we can find the circumcentre by drawing the perpendicular bisectors of sides \(AB, BC,\) and \(CA\) of this triangle. \(O\) is the point where these bisectors are meeting together. Therefore, \(O\) is the point which is equidistant from all the vertices of \(\Delta ABC\).

## Question 2

In a triangle locate a point in its interior which is equidistant from all the sides of the triangle.

### Solution

**Video Solution**

**Steps:**

The point which is equidistant from all the sides of a triangle is called the incentre of the triangle. Incentre of a triangle is the intersection point of the angular bisectors of the interior angles of that triangle.

Here, in \(\Delta ABC\), we can find the incentre of this triangle by drawing the angular bisectors of the interior angles of this triangle. **I** is the point where these angle bisectors are intersecting each other. Therefore, **I** is the point which is equidistant from all the sides of \(\Delta ABC\).

## Question 3

In a huge park people are concentrated at three points (see the given figure)

### Solution

**Video Solution**

A: where there are different slides and swings for children,

**B: **near which a man-made lake is situated,

**C:** which is near to a large parking and exit.

Where should an ice-cream parlour be set up so that maximum number of persons can approach it?

(*Hint:* The parlor should be equidistant from \(A, B\) and \(C\))

**Steps:**

Maximum number of persons can approach the ice-cream parlour if it is equidistant from \(A, B\) and \(C\). Now, \(A, B\) and \(C\) form a triangle. In a triangle, the circumcentre is the only point that is equidistant from its vertices. So, the ice-cream parlour should be set up at the circumcentre \(O\) of \(\Delta ABC\).

In this situation, maximum number of persons can approach it. We can find circumcentre \(O\) of this triangle by drawing perpendicular bisectors of the sides of this triangle.

## Question 4

Complete the hexagonal and star shaped rangolies (see the given figures) by filling them with as many equilateral triangles of side \(1\,\rm cm\) as you can. Count the number of triangles in each case. Which has more triangles?

### Solution

**Video Solution**

**Steps:**

It can be observed that hexagonal-shaped rangoli has 6 equilateral triangles in it.

Area of an equilateral Triangle=\( \frac { \sqrt { 3 } } { 4 } ( \text { side } ) ^ { 2 }\)

\[\begin{align}\text{Area of } \Delta OAB &= \frac { \sqrt { 3 } } { 4 } ( \text {side} )^{2}\\&= \frac { \sqrt { 3 } } { 4 } ( 25 ) \\ &= \frac { 25 \sqrt { 3 } } { 4 }\; cm ^ { 2 } \end{align}\]

Area of hexagonal-shaped Rangoli

\[\begin{align} &= 6 \times \frac { 25 \sqrt { 3 } } { 4 } \\ &= \frac { 75 \sqrt { 3 } } { 2 }\;cm ^ { 2 } \end{align}\]

Area of equilateral triangle having its side as \(1\,\rm cm\)

\[\begin{align} & = \frac { \sqrt { 3 } } { 4 } ( 1 ) ^ { 2 } \\ & = \frac { \sqrt { 3 } } { 4 } \; cm ^ { 2 } \end{align}\]

Number of equilateral triangles of 1 cm side that can be filled in this hexagonal-shaped Rangoli

\[\begin{align} & = \left( \frac { \frac { 75 \sqrt { 3 } } { 2 } } { \frac { \sqrt { 3 } } { 4 } } \right)\\ & =150 \end{align}\]

Star-shaped rangoli has \(12\) equilateral triangles of side \(5\,\rm cm\) in it.

Area of star-shaped rangoli

\[\begin{align}& = 12 \times \frac { \sqrt { 3 } } { 4 } \times ( 5 ) ^ { 2 } \\ & = 75 \sqrt { 3 } \end{align}\]

Number of equilateral triangles of 1 cm side that can be filled in this star-shaped rangoli

\[\begin{align} &= \left( \frac { 75 \sqrt { 3 } } { \frac { \sqrt { 3 } } { 4 } } \right) \\ &= 300 \end{align}\]

Therefore, star-shaped rangoli has more equilateral triangles in it.

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