Triangle inequality rule: The length of any side of a triangle will always be less than the sum of the lengths of the other two sides and greater than the difference of the lengths of the other two sides.
Sum of the interior angles: The sum of the angles of a triangle is 180 degrees.
Sum of the exterior angles: The sum of the angles of a triangle is 360 degrees.
Types of triangles:
Equilateral triangle:
Isosceles triangle: A triangle having two of its sides of equal length is known as isosceles triangle.
Scalene triangle: A triangle having three sides of different lengths is known as scalene triangle.
Acute triangle: A triangle having three acute angles is known as acute triangle. That is, if all three angles of a triangle are less than 90°, then it is an acute triangle.
Obtuse triangle:-
Right Angled triangle:-
Pythagorean theorem:-
In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
This is known as Pythagorean theorem.
Consider the below given right angle triangle,
Area of a triangle:
Consider a triangle with base of length \(b\) and height \(h\) as shown below.
The area of the triangle = \(\frac{1}{2}\) x \(b\) x \(h\).
Properties of triangles:
Theorems about congruency of triangles:
Two triangles are congruent, if they have accordingly equal:
i) Two sides and an angle between them,
ii) Two angles and a side, adjacent to them,
iii) all the three sides are equal.
Theorems about congruency of right-angled triangles:
Two right-angled triangles are congruent, if one of the following conditions is valid:
i) Their sides are equal,
ii) A side and a hypotenuse of one of triangle are equal to a leg and a hypotenuse of another.
ii) A hypotenuse and an acute angle of one of triangles are equal to a hypotenuse and acute angle of another.
Orthocenter:
Three heights (altitude) of triangle always intersect in one point, called an orthocenter of a triangle.
Altitude: It is a perpendicular, dropped from any vertex to an opposite side. this side is called a base of triangle.
Centroid:
Intersecting point of median is known as centroid as shown in the figure below. This point divides each median by ratio 2:1, considering from a vertex.
Median:
Median is a line segment, joining any vertex of triangle and a midpoint of the opposite side.
In-center:
Intersecting point of angle bisector is known as in-center.
Angle bisector:
It is a line segement from a vertex to a point of intersection with an opposite side.
Circumcenter:
Example 1:
The right triangle shown below has an area of 25. Find its hypotenuse.
Example 2:
Triangle ABC shown below is inscribed inside a square of side 20 cm. Find the area of the triangle
2. If AB = BD, BC = CD and \(\angle\)ACD = \({80}^{\circ}\), what is the measure in degrees of \(\angle\)BAD?
Solution:
3. In the figure given, DE\(\parallel\) BC. Find the length of DE?
4.
Quantity A | Quantity B |
Area of \(\bigtriangleup\)Q | 8 sq. m. |
A. The quantity on the left is greater.
B. The quantity on the right is greater.
C. Both are equal.
D. The relationship cannot be determined without further information.
Solution:
5.
Quantity A | Quantity B |
9 | \(x\) |
A. The quantity on the left is greater.
B. The quantity on the right is greater.
C. Both are equal.
D. The relationship cannot be determined without further information.
Solution: