ABC is a triangle right angled at C. Let p be the length of the perpendicular drawn from C on AB. If BC = 6 cm and CA = 8 cm, then what is the value of p?
4.8 cm
Understanding the Right-Angled Triangle Problem
We are dealing with a specific type of triangle: a right-angled triangle named ABC, with the right angle located at vertex C. The problem provides the lengths of the two sides that form the right angle: the side BC is 6 cm long, and the side CA is 8 cm long. Our task is to determine the length, denoted by the variable p, of the perpendicular line segment drawn from the right-angle vertex (C) down to the hypotenuse (the side AB).
Calculating the Hypotenuse Length
Before we can find the length of the perpendicular (p), we first need to know the length of the hypotenuse, AB. We can find this using the Pythagorean theorem, a fundamental rule in geometry for right-angled triangles.
The Pythagorean theorem states that the square of the hypotenuse's length is equal to the sum of the squares of the lengths of the other two sides.
The formula is:
\[AB^2 = BC^2 + CA^2\]
Let's plug in the given values for BC and CA:
\[AB^2 = (6 \text{ cm})^2 + (8 \text{ cm})^2\]
Calculate the squares:
\[AB^2 = 36 \text{ cm}^2 + 64 \text{ cm}^2\]
Add them together:
\[AB^2 = 100 \text{ cm}^2\]
To find the length of AB, we take the square root of both sides:
\[AB = \sqrt{100 \text{ cm}^2}\]
\[AB = 10 \text{ cm}\]
So, the hypotenuse AB measures 10 cm.
Calculating the Perpendicular Length (p) Using Area
A useful technique to find the length of the perpendicular (p) is to compare the area of the triangle calculated in two different ways:
- Method 1: Using the sides forming the right angle (BC and CA) as the base and height.
- Method 2: Using the hypotenuse (AB) as the base and the perpendicular (p) as the height.
Since both methods calculate the area of the same triangle, the results must be equal.
Area Calculation: Method 1
The standard formula for the area of a triangle is:
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \]
For triangle ABC, we can use BC as the base and CA as the height:
\[ \text{Area} = \frac{1}{2} \times BC \times CA \]
\[ \text{Area} = \frac{1}{2} \times 6 \text{ cm} \times 8 \text{ cm} \]
\[ \text{Area} = \frac{1}{2} \times 48 \text{ cm}^2 \]
\[ \text{Area} = 24 \text{ cm}^2 \]
Area Calculation: Method 2
Alternatively, we can use the hypotenuse AB as the base and p (the perpendicular from C to AB) as the height:
\[ \text{Area} = \frac{1}{2} \times AB \times p \]
We found that AB = 10 cm, so:
\[ \text{Area} = \frac{1}{2} \times 10 \text{ cm} \times p \]
\[ \text{Area} = 5p \text{ cm}^2 \]
Equating Areas to Find p
Now we set the two expressions for the area equal to each other:
\[ 5p \text{ cm}^2 = 24 \text{ cm}^2 \]
To solve for p, we divide both sides of the equation by 5:
\[ p = \frac{24 \text{ cm}^2}{5 \text{ cm}} \]
\[ p = 4.8 \text{ cm} \]
Conclusion
By using the properties of the right-angled triangle and calculating its area in two ways, we have found the length of the perpendicular drawn from vertex C to the hypotenuse AB.
The calculated value for p is 4.8 cm.
Similar Questions
What is the area of quadrilateral ABCD?
ABC is a triangle right angled at B. Let D be the midpoint on AC. If BD = 6.5 cm, then what is AB 2 + BC 2 equal to?
AD is the median of the triangle ABC. If P is any point on AD, then which one of the following is correct?
In a triangle ABC, if 2 ∠A = 3 ∠B = 6 ∠C, then what is ∠A + ∠C equal to?
Consider the following statements :
1. The sum of any two sides of a triangle is less than twice the median drawn to the third side.
2. The perimeter of a triangle is greater than the sum of the three medians.
Which of the above statements is/are correct?
In a triangle, values of all the angles are integers (in degree measure). Which one of the following cannot be the proportion of their measures?
ABC is an equilateral triangle. The side BC is trisected at D such that BC = 3 BD. What is the ratio of AD 2to AB 2?
Two isosceles triangles have equal vertical angles and their areas are in the ratio 4.84 ∶ 5.29. What is the ratio of their corresponding heights?
Δ ABC is similar to Δ DEF. The perimeters of Δ ABC and Δ DEF are 40 cm and 30 cm respectively. What is the ratio of (BC + CA) to (EF + FD) equal to?
What is the maximum number of circum-circles that a triangle can have?
Important Questions from Triangles, Congruence and Similarity
G is the centroid of the equilateral triangle ABC. If AB = 8√ 3 cm, then the length of AG is equal to:
If sides of a triangle are 12 cm, 15 cm and 21 cm, then what is the inradius (in cm) of the triangle?
What is the area of quadrilateral ABCD?
It is given that ΔABC ~ ΔXYZ and Area ΔABC : Area ΔXYZ = 81 : 25. If AB = 18 cm, BC = 10 cm, CA = 15 cm, then what is the side XZ (in cm)?
Sides of two similar triangles are in the ratio 4 ∶ 9. Area of these triangles are in the ratio:
