Triangle
Triangle is a three sided polygon i.e. a closed figure consisting of three line segments linked end-to-end. Vertex: The vertex is a corner of the triangle. Every triangle has three vertices. Base: The base of a triangle can be a side out of three sides, generally used is the one drawn at the bottom. Notice that in case of an isosceles triangle, the base is commonly taken to be the unequal side.
(Continued from above)
Altitude: The altitude of a triangle is the perpendicular from the base to the opposite vertex. It may be such that the base may need to be extended.
- As there are three bases in a triangle, there are three possible altitudes also.
- The three altitudes intersect at a single point, called the orthocenter of the triangle (O) in Triangle
ABC (Figure, on right). Median: The median of a triangle is a line from a vertex to the midpoint of the opposite side.
- The three medians intersect at a single point, called the centroid (O),
of the triangle triangle ABC (Figure, on right). It is the center of gravity of the triangle also.
- The median divides the triangle into two smaller triangles having the same area. The total number of small triangles shall be six.
- In any triangle type, the medians shall always intersect at a single point.
- Important to remember: two-thirds of the length of each median is between the vertex and the centroid, and one-third is between the centroid and the midpoint of the opposite side.
It is expressed as: here a,b,c represent the sides of the triangle and a is the side in the triangle whose midpoint is the extreme point of median m..
- Area of a triangle conceptually is the number of square units it takes to exactly fill the interior of a triangle. A general use formula is: A = 1/2 (b
* h) i.e. half of the base times the height.
Heron's for area of a triangle (given length of the three sides a, b and c of the triangle): A =. s represents (a + b + c)/2 i.e. one half of the perimeter.
Important Note: An equilateral triangle has the largest area, for a given perimeter, and for a given area equilateral triangle has the smallest perimeter.
Example 6: Given- a right triangle in the figure below, where the angles are 45°, 45°, and 90°. Find the area of Triangle ABC.
Recall the properties of special right triangle 45°, 45°, and 90°.
Step 1: It is known that the sides of this special triangle shall be in the ratio of 1:1:2. The longest side hypotenuse is 2. (Use Pythagorean Theorem).
Step 2: As the base angles measure same, i.e. both 45°the two legs are equal. Thus it is an isosceles triangle.
Step 3: Imagine two such triangles make square, with sides say. Thus area of
ABC shall be half of the area of square i.e.
s2 = 102 = 100. Thus area of the triangle is 50 cm2, as the final answer.
Example 7: Given- what is the area of the largest triangle that can be fitted into a rectangle of length 'l' units and width 'w' units?
Look at the figure on below and explore possible ways that triangle that can be fitted into a rectangle ABCD.
Step 1: It could be- the ABP which has its base as the length (l) and its height (w)
as the width of the rectangle. Other could be the
BCQ which has its base as the width (w) and height as the length of the rectangle (l) will be the largest triangle that can be fitted in the rectangle.
Step 2: If the base of the triangle is 'l' and its height 'w',
then its area A = 1/2 (lw) units. In other case if base of the triangle is 'w' and its height is 'l' units, then its area is 1/2 (lw) units.
Thus (lw)/2, is the final answer.
Example 8: Given- find out the measure of inradius of the triangle whose sides are 24, 7 and 25.
To have clear understanding on the given and what is required, draw the diagram. You can figure out this is case of a right triangle.
The inradius can be found by equating the area of the triangle ABC with the sum of the areas of the three triangles shown in the figure. The formula arrived at for radius r is:
r = (ab)/(a + b + c), where a,
b and c are the side lengths of the triangle. OR taking area A
of the triangle: A= r * s, where r is the radius and s is the semi perimeter of the triangle.
Step 1: A = 1 /2 * 24 * 7 = 84 sq units.
Step 2: s = (24 + 25 + 7)/2 = 28.
Step 3: r * 28 = 84.
Thus r = 3 units.
The inradius is 3 units, as the final answer.
Example 9: Given- find the area of the triangle whose vertices are (-6, -2), (-4, -6), (-2, 5).
Step 1: Draw the diagram to understand the problem
Step 2: The area of triangle as given by its vertices is:
= 1/2
= 1/2 (-x2y1 + x3y1 + x1y2 x3y2 x1y3 + x2y3) (1)
Where, (x1, y1), (x2, y2) and (x3, y3) are the vertices of the triangle.
Step 2: Given in the problem,
(x1, y1) = (-6, -2)
(x2, y2) = (-4, -6)
(x3, y3) = (-2, 5)
Step 3: Plug in values in (1) above, A = 1/2 (-(-4)(-2) + (-2)(-2) - (-2)(-6) - (-6)(5) + (-6)(-6) + (-4)(5))
= 1/2 (-8 + 4 12 + 36 + 30 - 20)
= 1/2 (-16 + 46)
= 1/2 (30)
= 15 sq units
Thus area of the given triangle is 15 sq units, as the final answer.
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here a,b,c represent the sides of the triangle and a is the side in the triangle whose midpoint is the extreme point of median m..
. s represents (a + b + c)/2 i.e. one half of the perimeter.
2. The longest side hypotenuse is 
ABC shall be half of the area of square i.e. 
