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SUBJECT:MATH
TOPIC....
CLASSIFICATION OF TRIANGLE.
We classify triangles with reference to the sides and angles.
On the basis of sides :(i) Scalene Triangle :
When all the sides of triangle are unequal, the triangle is called a scalene triangle
All angles of a scalene triangle are unequal. We can verify this by measurement (see figure).
(ii) Equilateral Triangle :
When all the three sides of a triangle are equal, the triangle is called an equilateral triangle (see figure).
is an equilateral triangle.
Further, all the angles of an equilateral triangle are equal. Each angle is equal to 60°.
(iii) Isosceles Triangle :
If two sides of a triangle are equal, it is called an isosceles triangle (see figure).
In
Side
is an isosceles triangle.
Further, angles opposite to the equal sides are also equal.
(see figure)
On the basis of angles :
(i) Acute-angled Triangle :
When all the angles of a triangle are acute, the triangle is called an acute-angled triangle. (See figure)
All angles of the AABC are acute i.e.,
are all acute angles.
(ii) Obtuse-angled Triangle :
When one angle of a triangle is an obtuse angle, the triangle is called obtuse-angled triangle (see figure).
In
is an obtuse angle.
So,
is an obtuse-angled triangle.
(iii) Right-angled Triangle :
When one angle of a triangle is a right angle, the triangle is called a right-angled triangle.
(see figure)
is a right-angled triangle with angle ABC as right angle (90°).
Illustrative Example
Example 1. Look at the figures and classify each of the triangle according to its (a) sides (b) angles.
Solution. (a) Classification of triangles according to their sides :
Scalene triangle : (ii)
Isosceles triangle : (i), (iii) and (v)
Equilateral triangle : (iv)
(b) Classification of triangles according to their angles :
Acute-angled triangle : (i) and (iv)
Right-angled triangle : (ii) and (v)
Obtuse-angled triangle : (iii)
MEDIANS OF A TRIANGLE
A median of a triangle is the line segment that joins a vertex to the mid-point of the opposite side.
Draw a median to a triangle.
Every triangle has three medians.
Steps of Construction :
Draw any triangle ABC.
On
B as centre and any radius more than half of
draw arcs on each side of
Now, with C as centre and the same radius draw arcs on each side of
cutting the earlier drawn arcs at X and Y.
Join XY meeting
onD. D is the mid-point of
Join
is the required median. Similarly, we can draw medians from B and C to
respectively.
ALTITUDES OF A TRIANGLE
An altitude of a triangle is the line segment from a vertex of the triangle perpendicular to the opposite side.
Every triangle has three altitudes, one from each vertex to the opposite side.
Draw an altitude to a triangle.
So,
is an obtuse-angled triangle.
(iii) Right-angled Triangle :
When one angle of a triangle is a right angle, the triangle is called a right-angled triangle.
(see figure)
is a right-angled triangle with angle ABC as right angle (90°).
Illustrative Example
Example 1. Look at the figures and classify each of the triangle according to its (a) sides (b) angles.
Solution. (a) Classification of triangles according to their sides :
Scalene triangle : (ii)
Isosceles triangle : (i), (iii) and (v)
Equilateral triangle : (iv)
(b) Classification of triangles according to their angles :
Acute-angled triangle : (i) and (iv)
Right-angled triangle : (ii) and (v)
Obtuse-angled triangle : (iii)
MEDIANS OF A TRIANGLE
A median of a triangle is the line segment that joins a vertex to the mid-point of the opposite side.
Draw a median to a triangle.
Every triangle has three medians.
Steps of Construction :
Draw any triangle ABC.
On
B as centre and any radius more than half of
draw arcs on each side of
Now, with C as centre and the same radius draw arcs on each side of
cutting the earlier drawn arcs at X and Y.
Join XY meeting
onD. D is the mid-point of
Join
is the required median. Similarly, we can draw medians from B and C to
respectively.
ALTITUDES OF A TRIANGLE
An altitude of a triangle is the line segment from a vertex of the triangle perpendicular to the opposite side.
Every triangle has three altitudes, one from each vertex to the opposite side.
Draw an altitude to a triangle.
Steps of Construction :
Draw a triangle ABC [acute-angled or obtuse-angled]
With A as centre and a suitable radius draw an arc cutting
produced (see figure (ii)) at two points X and Y.
Now, with X as centre and radius more than half XY draw an arc. Again, with Y as centre and the same radius draw another arc cutting the earlier drawn arc at Z.
Join AZ cutting the
at D.
is the required altitude.
Similarly, we can draw altitudes from other two vertices B and C to the line
respectively.
Illustrative Examples
Example 1. Draw rough sketches for the following :
Solution.
Example 2. Prove that the altitude bisects the base of an isosceles triangle.
Solution.
Example 3. Draw rough sketches of altitudes from A to
for the following triangles :
Solution. Rough sketches of altitudes from A to
for the given triangles are as under :
Draw a triangle ABC [acute-angled or obtuse-angled]
With A as centre and a suitable radius draw an arc cutting
produced (see figure (ii)) at two points X and Y.
Now, with X as centre and radius more than half XY draw an arc. Again, with Y as centre and the same radius draw another arc cutting the earlier drawn arc at Z.
Join AZ cutting the
at D.
is the required altitude.
Similarly, we can draw altitudes from other two vertices B and C to the line
respectively.
Illustrative Examples
Example 1. Draw rough sketches for the following :
Solution.
Example 2. Prove that the altitude bisects the base of an isosceles triangle.
Solution.
Example 3. Draw rough sketches of altitudes from A to
for the following triangles :
Solution. Rough sketches of altitudes from A to
for the given triangles are as under :
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