The sides of an equilateral triangle are(2p-q+5) and(2q-p+2)  If area of the triangle is root3/4.k  then k=
class 9 correct ans in number

Given the sides of an equilateral triangle as \( 2p - q + 5 \) and \( 2q - p + 2 \), we know that all sides of an equilateral triangle are equal. Therefore:

\[
2p - q + 5 = 2q - p + 2
\]

To solve for \( p \) and \( q \), first rearrange and simplify the equation:

\[
2p + p = 2q + q + 2 - 5
\]

\[
3p - 3q = -3
\]

\[
p - q = -1
\]

Now, substituting \( q = p + 1 \) into one of the side expressions, let’s take \( 2p - q + 5 \):

\[
2p - (p + 1) + 5 = p + 4
\]

So, each side of the equilateral triangle can be represented as \( p + 4 \).

The area \( A \) of an equilateral triangle with side \( a \) is given by:

\[
A = \frac{\sqrt{3}}{4} \times a^2
\]

Given that the area is \( \frac{\sqrt{3}}{4} \times k \):

\[
\frac{\sqrt{3}}{4} \times (p + 4)^2 = \frac{\sqrt{3}}{4} \times k
\]

This simplifies to:

\[
(p + 4)^2 = k
\]

Now, substitute \( p - q = -1 \), so \( q = p + 1 \):

\[
k = (p + 4)^2
\]

Thus, \( k \) is the square of one side length of the triangle, which is equal to \( (p + 4)^2 \).

So the correct numerical answer for \( k \) is \( (p + 4)^2 \).

If you have specific values of \( p \) and \( q \) or need further clarification, please provide them. 

Do you want further details or have any questions?

Here are some related questions you might find interesting:
1. What is the relationship between the perimeter and the area of an equilateral triangle?
2. How do we derive the formula for the area of an equilateral triangle?
3. How would the problem change if the triangle were not equilateral but isosceles?
4. What are the properties of equilateral triangles that distinguish them from other types of triangles?
5. How do you calculate the height of an equilateral triangle given its side length?

**Tip:** For equilateral triangles, all sides and angles are equal, making it easier to apply consistent geometric formulas.

Learn how to calculate the area of an equilateral triangle when given sides 2p - q + 5 and 2q - p + 2. Understand the geometric properties and apply the area formula for equilateral triangles. Suitable for Grade 9 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation