The sides of a triangle are in the ratio 6 : 4 : 3 and its perimeter is 104 cm. The length of the longest side (in cm) is:
48
Finding the Longest Side of a Triangle using Ratio and Perimeter
We are given a triangle where the lengths of its sides are in the ratio 6 : 4 : 3. The perimeter of the triangle is 104 cm. We need to find the length of the longest side.
Let the common ratio factor be \(x\). Then the lengths of the three sides of the triangle can be represented as \(6x\), \(4x\), and \(3x\).
The perimeter of a triangle is the sum of the lengths of its three sides. We are given that the perimeter is 104 cm.
So, we can write the equation for the perimeter:
\(6x + 4x + 3x = 104\)
Combine the terms on the left side:
\( (6 + 4 + 3)x = 104 \)
\( 13x = 104 \)
Now, solve for \(x\) by dividing both sides by 13:
\( x = \frac{104}{13} \)
\( x = 8 \)
Now that we have the value of \(x\), we can find the length of each side of the triangle.
- Side 1 (ratio 6): \(6x = 6 \times 8 = 48\) cm
- Side 2 (ratio 4): \(4x = 4 \times 8 = 32\) cm
- Side 3 (ratio 3): \(3x = 3 \times 8 = 24\) cm
The lengths of the sides are 48 cm, 32 cm, and 24 cm.
We need to find the length of the longest side. Comparing the three side lengths, we see that 48 cm is the largest value.
Therefore, the length of the longest side is 48 cm.
| Ratio | Side Length (\(x=8\)) |
|---|---|
| 6 | \(6 \times 8 = 48\) cm |
| 4 | \(4 \times 8 = 32\) cm |
| 3 | \(3 \times 8 = 24\) cm |
Revision Table: Triangle Ratio and Perimeter
Let's quickly review the key information and steps:
| Given Information | Calculation Step | Result |
|---|---|---|
| Side Ratio | 6 : 4 : 3 | |
| Perimeter | 104 cm | |
| Represent Sides | Set sides as \(6x\), \(4x\), \(3x\) | Total ratio sum = \(6+4+3 = 13\) |
| Perimeter Equation | \(13x = 104\) | |
| Solve for \(x\) | \(x = \frac{104}{13}\) | \(x = 8\) |
| Calculate Side Lengths | \(6 \times 8\), \(4 \times 8\), \(3 \times 8\) | 48 cm, 32 cm, 24 cm |
| Identify Longest Side | Compare 48, 32, 24 | 48 cm |
Additional Information on Triangle Properties
Understanding triangle properties is important for solving geometry problems. Here are a few related concepts:
- Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In our case, 24 + 32 > 48 (56 > 48), 24 + 48 > 32 (72 > 32), and 32 + 48 > 24 (80 > 24). This confirms that a triangle with these side lengths can exist.
- Types of Triangles: Triangles can be classified by their side lengths (Equilateral, Isosceles, Scalene) or angles (Acute, Right, Obtuse). A triangle with sides 48, 32, and 24 is a scalene triangle because all three sides have different lengths.
- Ratio and Proportion: Ratios are used to compare quantities. When solving problems involving ratios, introducing a common multiplier like \(x\) helps in setting up equations based on given information like perimeter or area.
Important Questions from Geometry
An isosceles right-angled triangle has hypotenuse length as 10 units. What is the area of the triangle (in square units)?
Two circles of radii 16 cm and 4 cm, respectively, touch each other externally at Point A. PQ is the direct common tangent of these circles with centres C1 and C2, respectively. What is the length of PQ?
Let C be a circle with center O and AB be a chord of C such that the length of AB is equal to the radius of C. Let D be any point on the major arc of AB. Find ∠AOB and ∠ADB, respectively.
The centres of two circles are 84 cm apart. If the radii of these two circles are 38 cm and 26 cm, respectively, then which of the following options gives the length (in cm) of a direct common tangent of these two circles?
In ADEF, the bisector of ∠D intersects side EF at point N. If DE = 36 cm, DF = 40 cm and EF = 38 cm, then the length (in cm) of NF is ________.
