Understanding the Triangle Sides
We are given a triangle with sides measuring 11 cm, 60 cm, and 61 cm. Our goal is to find the area of a new triangle formed by connecting the mid-points of the sides of this original triangle.
Checking for a Right-Angled Triangle
First, let's determine the type of triangle we have. We can use the Pythagorean theorem (\(a^2 + b^2 = c^2\)) to check if it's a right-angled triangle. Let's test if the square of the longest side is equal to the sum of the squares of the other two sides:
- Side 1 (\(a\)) = 11 cm
- Side 2 (\(b\)) = 60 cm
- Side 3 (\(c\)) = 61 cm
Calculate the squares:
- \(a^2 = 11^2 = 121\)
- \(b^2 = 60^2 = 3600\)
- \(c^2 = 61^2 = 3721\)
Now, check the theorem:
\(a^2 + b^2 = 121 + 3600 = 3721\)
Since \(a^2 + b^2 = c^2\) (\(3721 = 3721\)), the triangle is indeed a right-angled triangle. The sides 11 cm and 60 cm are the base and height (legs), and 61 cm is the hypotenuse.
Calculating the Area of the Original Triangle
The area of a right-angled triangle is calculated using the formula:
Area = \(\frac{1}{2} \times \text{base} \times \text{height}\)
Using the sides 11 cm and 60 cm as the base and height:
Area\(_{original}\) = \(\frac{1}{2} \times 11 \text{ cm} \times 60 \text{ cm}\)
Area\(_{original}\) = \(\frac{1}{2} \times 660 \text{ cm}^2\)
Area\(_{original}\) = \(330 \text{ cm}^2\)
Area of the Triangle Formed by Mid-points
There's a key property in geometry: the triangle formed by joining the mid-points of the sides of any triangle is similar to the original triangle and its area is exactly one-fourth (1/4) of the area of the original triangle.
Area\(_{mid-point}\) = \(\frac{1}{4} \times \text{Area}_{original}\)
Substituting the area we calculated:
Area\(_{mid-point}\) = \(\frac{1}{4} \times 330 \text{ cm}^2\)
Area\(_{mid-point}\) = \(82.5 \text{ cm}^2\)
Final Answer
Therefore, the area of the triangle formed by joining the mid-points of the sides of the given triangle is 82.5 cm\(^2\).
Similar Questions
What is \(AB + BC\) equal to?
- In a right-angled triangle ABC, AB = 15 cm, BC = 20 cm and AC = 25 cm. Further, BP is the perpendicular on AC. What is the difference in the area of triangles PAB and PCB?
Let X, Y and Z be the midpoints of the sides BC, CA and AB of a triangle ABC respectively. Consider the following statements:
I. The quadrilateral AZXY is a parallelogram.
II. The area of the quadrilateral AZXY is half of the area of the triangle ABC.
Which of the statements given above is/are correct?- In a triangle ABC, \(\angle A = 30^\circ\), AB = 7 cm and AC = 12 cm. What is the area of the triangle ABC?
- ABC is a triangle right angled at B. D is a point on AC such that BD is perpendicular to AC. If AB = \(p\) and BC = \(\sqrt{3}p\), then what is BD equal to?
Important Questions from Triangles
In a Δ ABC, if ∠A = 120° and AB = AC, then the values of ∠B and ∠C are respectively:
The product of the perimeter of a triangle, the radius of its in‐circle, and a number gives the area of the triangle. The number is
In ∆ABC, 2∠A = 3∠B = 6∠C. What is the value of the largest angle among these three angles?
The longest side in a right-angled triangle is called :
The sum of two sides of a triangle with the third side is
