All questions of The Triangles and its properties for Class 7 Exam

Yes i can the answer of this question is option D because in an isosceles triangle.two sides are equal in length

Exterior angle is equal to opposite interior angle so, when we add 50 degree to 30 it will come 80 degree .. that 'A' is correct

40

(c) right- angled

Because it is an equilateral triangle

Equilateral Triangle
An equilateral triangle is a type of triangle in which all three sides are of equal lengths. It is a special case of a regular polygon, where all sides and angles are equal. In an equilateral triangle, all three angles are also equal to 60 degrees.

Characteristics of an Equilateral Triangle
- All three sides of an equilateral triangle are of equal length.
- All three angles of an equilateral triangle are equal to 60 degrees.
- The sum of the interior angles of an equilateral triangle is always 180 degrees.
- The altitude, median, and perpendicular bisectors of an equilateral triangle are concurrent.
- The circumcenter, incenter, and centroid of an equilateral triangle coincide and are the same point.
- The area of an equilateral triangle can be calculated using various formulas, such as A = (s^2 * √3) / 4, where s is the length of a side.

Visual Representation
To visualize an equilateral triangle, imagine a triangle with all three sides of equal length. Each angle of the triangle measures 60 degrees. It appears symmetrical and balanced.

Distinguishing from Other Types of Triangles
- Scalene Triangle: In a scalene triangle, all three sides have different lengths. Therefore, it is not an equilateral triangle.
- Isosceles Triangle: In an isosceles triangle, two sides have the same length, while the third side is of a different length. Since all three sides of an equilateral triangle are equal, it is not an isosceles triangle either.

Importance of Equilateral Triangles
- Equilateral triangles are commonly used in architecture and engineering because of their symmetry and stability.
- They are often found in the design of bridges, buildings, and other structures.
- The properties of equilateral triangles are used in various mathematical and geometrical calculations.
- They are also used in various puzzles and games.

Conclusion
An equilateral triangle is a special type of triangle in which all three sides are of equal lengths. It possesses unique properties and characteristics that distinguish it from other types of triangles. Equilateral triangles have practical applications in various fields and are important in mathematical and geometrical concepts.

QR because in a right angled triangle hypotenuse is the biggest line and here point P is 90 degrees so the QR line would make hypotenuse

In a triangle the exterior angle is equal to the sum of its interior opposite angle, so 125 degree is equal to 90 degree+x. so 90+x=125 so.... x=125-90 then..... x=35 degree. I hope that it will help you and if you don't understand please don't hesitate me to ask it again........ .

Correct answer is option d because:-

Total perimeter of the triangle is 180 degree
= x+50+60=180
50+60=110
110-180
=70 degree

So please this is explained answer please followed.

please

Understanding the Triangle Inequality Theorem

The statement in the question refers to the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This theorem is a fundamental concept in geometry and is applicable to all types of triangles, including equilateral, isosceles, and scalene triangles.

Explanation of the Triangle Inequality Theorem

To understand why the sum of the lengths of any two sides is greater than the length of the third side, we need to consider the possible scenarios that can occur when constructing a triangle.

Scenario 1: The sum of the lengths of two sides is equal to the length of the third side
In this scenario, when the sum of the lengths of two sides of a triangle is equal to the length of the third side, the three sides are said to be in a straight line, and the triangle is called a degenerate triangle. In such a case, the triangle collapses into a line segment, and it is not considered a valid triangle.

Scenario 2: The sum of the lengths of two sides is less than the length of the third side
In this scenario, when the sum of the lengths of two sides is less than the length of the third side, it is impossible to construct a triangle. When we try to connect the two shorter sides, they will not meet to form a closed figure.

Scenario 3: The sum of the lengths of two sides is greater than the length of the third side
In this scenario, when the sum of the lengths of two sides is greater than the length of the third side, it is possible to construct a triangle. When we try to connect the two shorter sides, they will meet to form a closed figure, and all three sides will be connected.

Conclusion

Based on the Triangle Inequality Theorem, we can conclude that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. This theorem is essential in determining whether a given set of side lengths can form a valid triangle or not. Therefore, option 'A' - greater than, is the correct answer.

Understanding Exterior and Interior Angles
In geometry, the relationship between an exterior angle and its interior opposite angles is fundamental. The exterior angle is equal to the sum of the two interior opposite angles.
Given Information:
- One interior opposite angle (A) = 45 degrees
- Exterior angle (E) = 130 degrees
Finding the Second Interior Opposite Angle:
To find the second interior opposite angle (B), we can use the formula:
E = A + B
Substituting the known values:
130 = 45 + B
Now, isolate B:
B = 130 - 45
Calculating B:
- B = 130 - 45
- B = 85 degrees
Conclusion:
The measure of the second interior opposite angle is 85 degrees, which corresponds to option 'B'.
This relationship highlights how angles work together in geometry, reinforcing the concept of angle sums.

Understanding the Right-Angled Triangle
In a right-angled triangle, if the two legs are equal, we can refer to them as 'x'. According to the Pythagorean theorem, the relationship between the legs and the hypotenuse is given by:
- Hypotenuse² = Leg₁² + Leg₂²
Since both legs are equal:
- Hypotenuse² = x² + x²
- Hypotenuse² = 2x²
Given Information
We know that the square of the hypotenuse is 100 square units:
- 2x² = 100
Solving for x
To find the length of each leg, follow these steps:
1. Divide both sides by 2:
- x² = 100 / 2
- x² = 50
2. Now, take the square root of both sides to find 'x':
- x = √50
- x = √(25 * 2)
- x = 5√2
Conclusion
Thus, the length of each leg of the triangle is 5√2 units. Therefore, the correct answer is:
Option 'B': 5√2 units