NCERT Revision Notes for Chapter 6 Triangles Class 10 Maths
CBSE NCERT Revision Notes1
Answer
Two geometrical figures are said to be similar figures, if they have same shape but not necessarily the same size.
Or
A shape is said to be similar to other, if the ratio of their corresponding sides is equal and the corresponding angles are equal.
• Two polygons having the same number of sides are similar, if:
(i) all the corresponding angles are equal and
(ii) all the corresponding sides are in the same ratio or proportion.
If only one condition from (i) and (ii) is true for two polygons, then they cannot be similar.
If only one condition from (i) and (ii) is true for two polygons, then they cannot be similar.
• The ratio that compares the measurements of two similar shapes, is called the scale factor or representative fraction. It is equal to the ratio of corresponding sides of two figures.
We can use the ratio of corresponding sides to find unknown sides of similar shapes.
2
Answer
Two triangles are said to be similar triangles, if their corresponding angles are equal and their corresponding sides are proportional (i.e., the ratios between the lengths of corresponding sides are equal). For example: If in ∆ABC and ∆PQR
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and,
AB/PQ = BC/QR = AC/PR
The, △ABC∼△PQR
where, symbol ∼ is read as, ‘is similar to’.
Conversely
If △ABC is similar to △PQR, then
∠A = ∠P, ∠B = ∠Q, ∠C = ∠R and,
AB/PQ = BC/QR = AC/PR
Note: The ratio of any two corresponding sides in two equiangular triangles is always the same.
3
Answer
If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then other two sides are divided in the same ratio.
• The line segment joining the mid-points of any two sides of a triangle is parallel to the third side. (Mid-point theorem)
• The internal bisector of an angle of a triangle divides the opposite side internally in the ratio of the sides containing the angle. (Angle bisector theorem)
4
Answer
If a line divides any two sides of a triangle in the same ratio, then the line must be parallel to the third side (converse of basic proportionality theorem).
5
Answer
(i) In two triangles, if corresponding angles are equal, then their corresponding sides are in the same ratio i.e., they are proportional and hence the two triangles are similar. This AA Criteria for similarity of two triangles.
∠A=∠D
∠B=∠E
∠C=∠F
∆ABC~∆DEF by AAA
(ii) If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar (because by the angle sum property of a triangle, their third angle will also be equal) and it is called AA similarity.
6
Answer
7
Answer
8
Answer
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
If ∆ABC and ∆PQR are similar, then
ar (∆ABC)/ar (∆PQR) = (AB/PQ)2 = (AC/PR)2 =(BC/QR)2
9
Answer
(i) In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. (Pythagoras theorem)
(ii) In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite to first side is a right angle. (Converse of Pythagoras theorem)