In this chapter, we shall learn more about the topic, ‘Properties Of Triangle’ for Class 7. The triangle is a familiar geometric figure. Various shapes, such as a pyramid or even a hut, are triangular. Get ready to explore this important figure.
A triangle is an enclosed three-sided figure. All its three sides are adjacent or side-by-side. There are three angles contained within these three sides. Hence, there are no pairs of opposite sides in a triangle.
The sum or addition of all angles in a triangle equals one hundred eighty degrees. Recall the three types of angles. An acute angle measures between zero and ninety degrees. An obtuse angle measures between ninety and 180 degrees. And a right angle is precisely ninety degrees. Triangles are also varied based on the measurement of the angle.
A triangle can have a maximum of three acute angles, and should have a minimum of two acute angles. It also cannot have more than one right angle or one obtuse angle, because the total of all angles should not be more than 180 degrees. This measure is constant in every triangle, which we will see further. You may experiment with this change in condition to see if the angles add up to this constant of 180 degrees.
Hence, as the measure of one of the angles increases, the measure of the other two angles decreases. Similarly, as one of the angles decreases in measure, the other angles increase in measure. For class 7, the properties of a triangle are as follows:
A median is a line segment extended from a vertex to the opposite side in a triangle. It meets the opposite side at the midpoint. A median bisects the inner area of a triangle into two equal parts.There can thus be three medians in any triangle.
Make a paper triangle, and then join one of the vertices with the midpoint of the opposite side. A segment is formed. Now fold the paper at the segment. The paper folds exactly in half.
The point of intersection of the three medians is called the point of concurrence. This point is the exact midpoint of the inner area of the triangle. Take a triangle made out of cardboard, place it on a pencil tip exactly at this point. It will remain balanced in the air just on this tip.
In the MSVgo learning app, there are learning videos with NCERT solutions to watch and interactively learn as in the above example.
The altitude is also called the height of the triangle. It is the segment that is perpendicular to the opposite side from one vertex. There can be three altitudes for any triangle.
The altitudes intersect at the orthocenter. In an acute angle triangle, the orthocenter is inside the triangle. It is outside for an obtuse angle triangle. The sides of the triangle with an obtuse angle can be extended further so that the altitude can be constructed on them. The sides making ninety degrees in a right-angled triangle are already two of its altitudes.
Regular study helps prepare well for exams, understand concepts easily and solve any math problems quickly.
Do you recall reading about the property where the total sum of a triangle is 180 degrees? The angle measured on a straight line is also 180 degrees. If one of the sides of a triangle is extended outwards, the angle made on the outside is called a remote exterior angle.
Construct a triangle ABC, extend side CB outwards to a point D. The two angles on the segment CD at vertex B are angle ABC and angle ABD.
Angles ABC + ABD = 180 degrees
Angles A + ABC + C = 180 degrees
Angle ABD = Angles A + Angle C = 180 degree – Angle ABC
Angle ABD is the exterior angle, measuring equal to angles A and C.
Therefore, an exterior angle of a triangle is equal to the sum of its interior opposite angles.
Based on the relation between the angles, as well as the sides, there are three types of triangles:
Scalene Triangles: There are no sides of these triangles with equal measure. Hence, the angles do not have equal measure either.
Now for the special triangles:
Equilateral Triangles: In this triangle, all sides are equal to each other and all angles are equal to each other. Since the total sum of all angles must be 180 degrees, all three angles in every equilateral triangle must be measuring 60 degrees.
In this special triangle, the medians are also the altitudes. Hence, the point of concurrence and the orthocenter are the same in an equilateral triangle.
Isosceles Triangles: These triangles have two of their sides equal in length. The base angles that are opposite to the equal sides are also equal.
For example: In an isosceles triangle ABC, angle B = angle C; side AB = side AC. Angle A has a different measure. Side CB measures differently too.
The sum of the lengths of any two sides of a triangle is greater than the third side. Take three spots – the home (A), the shop (B) and the school (C). Joining these three spots makes a triangle ABC. Which distance will be the shortest from point A to point B? Will it be the direct route AB, or the route ACB?
The shortest distance between two points is a straight line. Hence, line segment AB will be the shorter route than AC + CB. This proves that the sum of any two sides is greater than its third side.
To be exam ready and also improve maths skills, it is important to have quick revisions of such topics day-to-day.
A triangle with one of the angles measuring 90 degrees is referred to as a right-angled triangle. The side opposite to the right angle is called the hypotenuse. The other two sides making the right angle are called legs of the triangle. Or, the perpendicular and the base.
The Indian mathematician Baudhayan and the Greek, Pythagoras discovered a rule about right-angled triangles:
In a right-angled triangle ABC, if AC is the hypotenuse, then, (AC)2 = (AB)2 + (CB)2
The converse of this theorem is also true. The converse says that if the square of the longest side of a triangle is equal to the sum of squares of the other two sides, the triangle must be a right-angled triangle.
If an isosceles triangle is right-angled, the pair of equal angles will both be 45 degrees. The sum of all angles is 180 degrees. One angle measuring 90 degrees, the sum of the other two angles must be 90 degrees. Hence, all three angles will make 180 degrees.
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What are the important topics to learn in the NCERT Solutions for Class 7 Maths Triangles And Its Properties?
These are the concepts presented in the NCERT Solutions for Class 7 Maths Triangles And Its Properties:
1. Medians of a Triangle
2. Altitudes of a Triangle
3. Exterior Angle of a Triangle and Its Property
4. Angle Sum Property of a Triangle
5. Equilateral and Isosceles Triangles: Two Special Triangles
6. Sum of the Lengths of Two Sides of A Triangle
7. Right-Angled Triangles and the Pythagoras Property
2. What are the exercises in the NCERT chapter for Class 7 Maths Triangles And Its Properties?
The exercises are based on the seven topics mentioned above. They follow after each subsection and test the student’s understanding of that particular section. This ensures a thorough grasp of the chapter.
3. Does MSVgo offer solutions for NCERT Class 7 Maths Triangles and Its Properties?
Yes. The MSVgo website and learning app offer solutions for questions in the textbook and simplified, crisp explanations of the chapters.